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+{"text":"\\section{Introduction and physics motivations} \\label{0}\n\nIn the present paper we compute, to the lowest perturbative order in $SU(N)$ Yang-Mills (YM) theory, $n$-point connected correlators, $G^{(n)}_{conf}(x_1,\\ldots,x_n)$, in the coordinate representation of the gauge-invariant twist-$2$ operators with maximal spin along the $p_+$ direction, both in Minkowskian and -- by analytic continuation -- Euclidean space-time. \\par\nIn fact, our computation matches and extends the previous lowest-order perturbative computation of $2$- and $3$-point gluonic correlators of twist-$2$ operators in $\\mathcal{N}=4$ SUSY YM theory \\cite{Kazakov:2012ar}, by including the unbalanced  \\footnote{In our terminology 'unbalanced' and 'balanced' refers to either the different or  the equal number of dotted and undotted indices that the aforementioned operators possess in the spinorial representation respectively. Unbalanced operators are referred to as 'asymmetric' in \\cite{Beisert:2004fv} and 'anisotropic' in \\cite{Robertson:1990bf}.} operators with collinear twist $2$ in pure YM theory and, most importantly, by calculating all the $n$-point correlators in the balanced and unbalanced sectors separately, and the $3$-point correlators in the mixed sector as well.\nOur physics motivation is threefold: \\par\nFirstly, our lowest-order computation has an intrinsic interest in YM theory, and -- according to \\cite{Kazakov:2012ar} -- in theories that extends it, such as its supersymmetric versions and QCD.\\par\nSecondly, our computation is preliminary to work out the ultraviolet (UV) asymptotics \\cite{Bochicchio:2013tfa, Bochicchio:2013eda} -- based on the renormalization-group (RG) improvement of perturbation theory -- of the above Euclidean $n$-point correlators.\\par\nThirdly, our computation is an essential ingredient to test the prediction in section $3$ of \\cite{Bochicchio:2016toi} that, by fundamental principles of the large-$N$ 't Hooft expansion, the generating functional of the nonperturbative leading nonplanar contributions to the aforementioned Euclidean correlators must have the structure of the log of a functional determinant \\cite{Bochicchio:2016toi} that sums the glueball one-loop diagrams.\\par\nIndeed, according to the philosophy of the asymptotically free bootstrap outlined in \\cite{Bochicchio:2016toi}, the RG-improved correlators mentioned above must be asymptotic in the UV \\cite{Bochicchio:2016toi}\nto the corresponding nonperturbative correlators involving glueballs. Therefore, to the leading nonplanar order, the generating functional of the former must share with the one of the latter the very same structure of the log of a functional determinant. \\par\nHence, our computation is the first step in both the directions mentioned above. \\par\n\n\n\\section{Main results} \\label{00}\n\n\n\\subsection{Balanced and unbalanced twist-$2$ conformal operators}\n\nWe describe our calculation and the operators that enter it. We compute, to the lowest perturbative order in $SU(N)$ YM theory, $n$-point connected correlators in Minkowskian space-time of the gauge-invariant twist-$2$ operators with maximal spin along the $p_+$ direction:\n\\begin{equation}\n\\langle \\mathcal{O}_1(x_1)\\ldots\\mathcal{O}_n(x_n)\\rangle_{lowest \\, order} = G^{(n)}_{conf}(x_1,\\ldots,x_n)\n\\end{equation}\nTo the lowest order, and to the next one \\footnote{In fact, to the order of $g^2$, in the conformal subtraction scheme \\cite{Braun:2003rp}.}, YM theory is conformal invariant \\cite{Braun:2003rp},\nsince the beta function only affects the solution of the Callan-Symanzik equation starting from the order of $g^4$. \nTherefore, following \\cite{Ohrndorf:1981qv, Braun:2003rp}, we employ operators that have nice transformation properties with respect to the collinear conformal subgroup involving the coordinate $x^+$.\\par\nPrimary conformal operators $\\mathcal{O}_{j}(x)$, with collinear conformal spin $j= s+\\frac{\\tau}{2}$, where $\\tau$ is the collinear twist and $s$ the collinear spin, i.e., the spin projected along the $p_+$ direction, transform under the action of the generators \\cite{Braun:2003rp} of the collinear conformal algebra $SL(2,\\mathbb{R})$:\n\\begin{align}\n&\\left[L_0,L_\\mp\\right]=\\mp L_\\mp  \\\\\\nonumber\n&\\left[L_-,L_+\\right]=-2L_0\n\\end{align}\naccording \\cite{Braun:2003rp} to:\n\\begin{align}\n\\label{confcom}\n&\\left[L_+,\\mathcal{O}_j(x)\\right] = -\\partial_{+}\\mathcal{O}_j(x)\\\\\\nonumber\n&\\left[L_-,\\mathcal{O}_j(x)\\right] = (x^{+\\,2}\\partial_++2jx^+)\\mathcal{O}_j(x)\\\\\\nonumber\n&\\left[L_0,\\mathcal{O}_j(x)\\right] = (x^+ \\partial_++j)\\mathcal{O}_j(x)\n\\end{align}\nwhere in eq. \\eqref{confcom} $x=(x^+,x^-,x^1,x^2)$ is restricted \\cite{Braun:2003rp} to the line $x^-=x^1=x^2=0$. Their conformal descendants, $\\partial_+^i \\mathcal{O}_j(x)$, are obtained by taking derivatives with respect to $x^+$, and have the same $\\tau$.\\par  For a given canonical dimension $d=\\tau+s$, the quasi-partonic \\cite{BUKHVOSTOV1985601} operators have minimum $\\tau$ and maximum $s$, with nice mixing (under renormalization) and conformal properties as above \\cite{Ohrndorf:1981qv, BUKHVOSTOV1985601, Braun:2003rp, Belitsky:2003sh, Belitsky:2004sc, Braun:2008ia}.\\par\nTheir collinear twist $\\tau$ does not necessarily coincide  \\cite{Braun:2003rp} with the twist $\\mathcal T$ -- defined by $d=\\mathcal T+S$, where $S$ is the spin -- that refers to the conformal group instead of the collinear subgroup.\\par\nAn infinite family of quasi-partonic operators is constructed as follows.\nA composite gauge-covariant primary conformal operator, built by two elementary \\footnote{In the present paper, we refer to the operators $\\Phi_{j}(x)$ as elementary, since they play the role of elementary constituents, though they may actually be composite operators.}\ngauge-covariant primary conformal operators $\\Phi_{j_1},\\Phi_{j_2}$, with collinear conformal spins $j_1, j_2$, has the form \\cite{Braun:2003rp}:\n\\begin{equation}\n\\label{maximum2}\n\\mathbb{O}_l^{j_1 j_2}(x) = \\Phi_{j_1}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^l P_l^{(2j_1-1,2j_2-1)}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)\\Phi_{j_2}(x)\n\\end{equation}\nwhere $P_l^{(2j_1-1,2j_2-1)}$ are Jacobi polynomials (appendix \\ref{appB}), $D_+$ is the covariant derivative along the $p_+$ direction (appendix \\ref{appN}), and the arrows denote the action of the derivative on the right or the left. The corresponding gauge-invariant object is obtained by taking the color trace.\\par\nThe collinear conformal spin, $j$, of the operator, $\\mathbb{O}_l^{j_1 j_2}(x)$, is $j =j_1+j_2+l$, where $l$ is the power of the derivative in eq. \\eqref{maximum2}.\nBy working out the definition in eq \\eqref{maximum2}, we get:\n\\begin{align}\n\\nonumber\n\\mathbb{O}_l^{j_1j_2}(x) \n&=\\sum_{k = 0}^{l} {l+2j_1-1\\choose k}{l+2j_2-1\\choose k+2j_2-1}(-1)^{l-k} \\Phi_{j_1}(x) \\overleftarrow{D}_+^{l-k} \\overrightarrow{D}_+^k\\Phi_{j_2}(x)\\\\\n&=\\sum_{k = 0}^{l} \\mathbb{O}_{l k}^{j_1 j_2}(x)\n\\end{align}\nthus realizing the conformal operator $\\mathbb{O}_l^{j_1j_2}(x)$ as a sum of $l+1$ operators, $\\mathbb{O}_{l k}^{j_1 j_2}(x)$, that are not necessarily conformal.\\par\nHence, the composite operators depend on a choice of the elementary conformal operators $\\Phi_{j_1},\\Phi_{j_2}$.\nWe define the standard conformal basis for primary operators with collinear twist $2$, where the elementary operators are $f_{11},f_{\\dot{1}\\dot{1}}$ (section \\ref{standardb}) with conformal spin $j=\\frac{3}{2}$. In the standard basis the gluonic operators are classified as in \\cite{Beisert:2004fv}:\n\\begin{align} \\label{OO}\n\\nonumber\n&\\mathbb{O}_{s} = \\Tr f_{11}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right) f_{\\dot{1}\\dot{1}}(x) \\qquad s = 2,4,6,\\ldots  \\\\\\nonumber\n&\\tilde{\\mathbb{O}}_{s} = \\Tr f_{11}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right) f_{\\dot{1}\\dot{1}}(x) \\qquad s = 3,5,7,\\ldots  \\\\\\nonumber\n&\\mathbb{S}_{s} =\\frac{1}{\\sqrt{2}}\\Tr f_{11}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)f_{11}(x) \\qquad s = 2,4,6,\\ldots\\\\\n&\\bar{\\mathbb{S}}_{s} =\\frac{1}{\\sqrt{2}}\\Tr f_{\\dot{1}\\dot{1}}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right) f_{\\dot{1}\\dot{1}}(x)\\qquad s = 2,4,6,\\ldots\n\\end{align}\nwith $C^{\\alpha}_{l}$ Gegenbauer polynomials (appendix \\ref{appB}), which are a special case of Jacobi polynomials.\\par\n$\\mathbb{O}_{s}$ and $\\tilde{\\mathbb{O}}_s$ are Hermitian balanced operators with $\\tau=\\mathcal T=2$. They have an equal number of undotted and dotted spinor indices (appendices \\ref{appA200} and \\ref{appA2}):\n\\begin{align}\n\\nonumber\n&\\mathbb{O}_s = \\mathbb{O}_{1\\dot{1}\\ldots 1\\dot{1}}\\\\\n&\\tilde{\\mathbb{O}}_s = \\tilde{\\mathbb{O}}_{1\\dot{1} \\ldots \\dot{1} 1}\n\\end{align}\n$\\mathbb{S}_{s}$ and its Hermitian conjugate, $\\bar{\\mathbb{S}}_{s}$, denoted by the bar superscript, are unbalanced operators with $\\tau=2$. \nThey have a different number of undotted and dotted spinor indices:\n\\begin{align}\n\\nonumber\n&\\mathbb{S}_s = \\mathbb{S}_{1111\\ldots 1\\dot{1}}\\\\\n&\\bar{\\mathbb{S}}_s= \\mathbb{S}_{\\dot{1}\\dot{1}\\dot{1}\\dot{1} \\ldots \\dot{1} 1}\n\\end{align}\nBesides, we also define the extended conformal basis for primary operators with collinear twist $2$, where the elementary operators are $D_{+}^{-1}f_{11}, D_{+}^{-1}f_{\\dot{1}\\dot{1}}$, with conformal spin $j=\\frac{1}{2}$, which are nonlocal in general, but local (appendix \\ref{appC}) in the light-cone gauge $A_+=0$. Clearly, gauge invariance ensures that all their correlators are local, as we verify explicitly. \\par\nThe extended basis is natural in SUSY calculations \\cite{Belitsky:2004sc}, and includes (nonlocal) operators with $\\tau=2$ and $s=0,1$. We have chosen it in YM theory because of the simplicity of the results for the correlators.\nIn the extended basis (section \\ref{nonstandardb}) the gluonic operators are:\n\\begin{align} \\label{AA}\n\\nonumber\n&\\mathbb{A}_{s} = \\Tr D_{+}^{-1}f_{11}(x)\\,(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)D_{+}^{-1} f_{\\dot{1}\\dot{1}}(x)  \\qquad s = 0,2,4,\\ldots\\\\\\nonumber\n&\\tilde{\\mathbb{A}}_{s} = \\Tr D_{+}^{-1}f_{11}(x)\\,(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right) D_{+}^{-1} f_{\\dot{1}\\dot{1}}(x)  \\qquad s = 1,3,5,\\ldots\\\\\\nonumber\n&\\mathbb{B}_{s} =\\frac{1}{\\sqrt{2}}\\Tr D_{+}^{-1}f_{11}(x)\\,(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)D_{+}^{-1}f_{11}(x)\\qquad s = 0,2,4,\\ldots\\\\\n&\\bar{\\mathbb{B}}_{s} =\\frac{1}{\\sqrt{2}}\\Tr D_{+}^{-1} f_{\\dot{1}\\dot{1}}(x)\\,(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)D_{+}^{-1} f_{\\dot{1}\\dot{1}}(x) \\qquad s = 0,2,4,\\ldots\n\\end{align}\n\n\n\\subsection{Minkowskian $n$-point correlators}\n\n\\subsubsection{Standard basis}\n\nWe have normalized our operators in such a way that the $2$-point correlators in the standard basis are equal for even $s$:\n\\begin{align}\n&\\langle {\\mathbb{O}}_{s_1}(x) {\\mathbb{O}}_{s_2}(y)\\rangle = \n\\langle \\mathbb{S}_{s_1}(x)\\bar{\\mathbb{S}}_{s_2}(y)\\rangle  =  \\mathcal{C}_{s_1}(x,y) \\delta_{s_1 s_2} \n\\end{align}\nand for odd $s$:\n\\begin{align}\n\\langle \\tilde{\\mathbb{O}} _{s_1}(x)\\tilde{\\mathbb{O}} _{s_2}(y)\\rangle =  \\mathcal{C}_{s_1}(x,y) \\delta_{s_1 s_2}\n\\end{align}\nwith:\n\\begin{align}\n\\nonumber\n\\mathcal{C}_{s}(x,y) =&\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{2s+2}i^{2s-4}}{(4!)^2}(s+1)^2(s+2)^2\n\\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s-2}\\sum_{k_2 = 0}^{s-2}{s\\choose k_1}{s\\choose k_1+2}{s\\choose k_2}{s\\choose k_2+2}(-1)^{s-k_2+k_1}\\\\\\nonumber\n&(s-k_1+k_2)!(s+k_1-k_2)! \\\\\\nonumber\n=&\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4}\\frac{2^{2s+2}i^{2s-4}}{(4!)^2}(s+1)^2(s+2)^2\n (2s)! \\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\\\\\n&\\sum_{k_1 = 0}^{s-2}\\sum_{k_2 = 0}^{s-2}{s\\choose k_1}{s\\choose k_1+2}{s\\choose k_2}{s\\choose k_2+2}(-1)^{k_2+k_1} \\frac{1}{{2s\\choose k_1+k_2+2}}\n\\end{align}\nwhere we omit the $i\\epsilon$ prescription in the propagators in the coordinate representation, in such a way that (appendix \\ref{appN}): \n\\begin{equation}\n\\frac{1}{\\rvert x\\rvert^2} \n\\end{equation}\nshould be read (appendix \\ref{appA1}):\n\\begin{equation}\n\\frac{1}{\\rvert x\\rvert^2-i\\epsilon}\n\\end{equation}\nThe very same correlators are evaluated by a trick \\cite{Kazakov:2012ar} (appendix \\ref{appA3}):\n\\begin{align}\n\\label{c2intro}\n\\mathcal{C}_{s}(x,y) = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{2s+2}}{(4!)^2}(-1)^s(s-1)s(s+1)(s+2)(2s)!\n\\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\n\\end{align}\nTherefore, we have discovered the following -- seemingly nontrivial -- identity (section \\ref{coordinates}):\n\\begin{align}\n\\frac{s_1(s_1-1)}{(s_1+1)(s_1+2)}\\delta_{s_1 s_2}=\n\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}(-1)^{k_2+k_1}\\frac{1}{{s_1+s_2\\choose k_1+k_2+2}}\n\\end{align}  \nWe have not found a direct proof of the above identity, but we have verified it numerically. \\par\nMoreover, the only nonvanishing $3$-point correlators are:\n\\begin{align}\n\\label{defc3}\n\\langle {\\mathbb{O}}_{s_1}(x){\\mathbb{O}}_{s_2}(y){\\mathbb{O}}_{s_3}(z)\\rangle=\\langle {\\mathbb{O}}_{s_1}(x){\\mathbb{S}}_{s_2}(y)\\bar{\\mathbb{S}}_{s_3}(z)\\rangle  = \\mathcal{C}_{s_1s_2s_3}(x,y,z)\n\\end{align}\nand:\n\\begin{align}\n\\label{defc3}\n\\langle {\\mathbb{O}}_{s_1}(x)\\tilde{\\mathbb{O}}_{s_2}(y)\\tilde{\\mathbb{O}}_{s_3}(z)\\rangle = \\mathcal{C}_{s_1s_2s_3}(x,y,z)\n\\end{align}\nwith:\n\\begin{align} \\label{312}\n\\nonumber\n \\mathcal{C}_{s_1s_2s_3}(x,y,z) \n=&-\\frac{1}{(4\\pi^2)^3} (1+(-1)^{s_1+s_2+s_3})   \\left(\\frac{2}{4!}\\right)^3\\frac{N^2-1}{8}i^{s_1+s_2+s_3}2^{s_1+s_2+s_3}\\\\\\nonumber\n& (s_1+1)(s_1+2)(s_2+1)(s_2+2)(s_3+1)(s_3+2)\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}\\sum_{k_3 = 0}^{s_3-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}{s_3\\choose k_3}{s_3\\choose k_3+2}\\\\\\nonumber\n&(s_1-k_1+k_2)!(s_2-k_2+k_3)!(s_3-k_3+k_1)!  \\\\\n&\\frac{(x-y)^{s_1-k_1+k_2}_+}{(\\rvert x-y\\rvert^2)^{s_1+1-k_1+k_2}}\\frac{(y-z)^{s_2-k_2+k_3}_+}{(\\rvert y-z\\rvert^2)^{s_2+1-k_2+k_3}}\\frac{(z-x)^{s_3-k_3+k_1}_+}{(\\rvert z-x\\rvert^2)^{s_3+1-k_3+k_1}}\n\\end{align}\nWe also compute the $n$-point correlators. In the balanced sector, we get:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{O}_{s_1}(x_1)\\ldots \\mathbb{O}_{s_n}(x_n)\\rangle_{conn}=\\frac{1}{(4\\pi^2)^n}\\frac{N^2-1}{2^n}2^{\\sum_{l=1}^n s_l}i^{\\sum_{l=1}^n s_l} \\\\\\nonumber\n&\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma(s_n+3)}{\\Gamma(5)\\Gamma(s_n+1)}\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_n = 0}^{s_n-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots {s_n\\choose k_n}{s_n\\choose k_n+2}\\\\\\nonumber\n&\\frac{(-1)^n}{n}\\sum_{\\sigma\\in P_n}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n)}-x_{\\sigma(1)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\nThe very same formula holds for an even number of operators $\\tilde{\\mathbb{O}}_s$, otherwise the correlators vanish.\nThe nonvanishing correlators in the balanced sector are:\n\\begin{align} \\label{O}\n\\nonumber\n&\\langle \\mathbb{O}_{s_1}(x_1)\\ldots \\mathbb{O}_{s_n}(x_n)\\tilde{\\mathbb{O}}_{s_{n+1}}(x_{n+1})\\ldots \\tilde{\\mathbb{O}}_{s_{n+2m}}(x_{n+2m})\\rangle_{conn}\\\\\\nonumber\n& =\\frac{1}{(4\\pi^2)^{n+2m}}\\frac{N^2-1}{2^{n+2m}}2^{\\sum_{l=1}^{n+2m} s_l}i^{\\sum_{l=1}^{n+2m} s_l}\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)} \\ldots\\frac{\\Gamma(3)\\Gamma(s_{n+2m}+3)}{\\Gamma(5)\\Gamma(s_{n+2m}+1)}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_{n+2m} = 0}^{s_{n+2m}-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots{s_{n+2m}\\choose k_{n+2m}}{s_{n+2m}\\choose k_{n+2m}+2}\\\\\\nonumber\n&\\frac{(-1)^{n+2m}}{n+{2m}}\\sum_{\\sigma\\in P_{n+2m}}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n+2m)}-x_{\\sigma(1)})_+^{s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n+{2m})}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)}+1}}\\,\n\\end{align}\nIn the unbalanced sector, we get:\n\\begin{align} \\label{S}\n\\nonumber\n&\\langle \\mathbb{S}_{s_1}(x_1)\\ldots \\mathbb{S}_{s_n}(x_n)\\bar{\\mathbb{S}}_{{s'\\!\\!}_1}(y_1)\\ldots \\bar{\\mathbb{S}}_{{s'\\!\\!}_n}(y_n)\\rangle=\\frac{1}{(4\\pi^2)^{2n}}\\frac{N^2-1}{2^{2n}}2^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}i^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}\\\\\\nonumber\n&\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma(s_n+3)}{\\Gamma(5)\\Gamma(s_n+1)}\\frac{\\Gamma(3)\\Gamma({s'\\!\\!}_1+3)}{\\Gamma(5)\\Gamma({s'\\!\\!}_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma({s'\\!\\!}_n+3)}{\\Gamma(5)\\Gamma({s'\\!\\!}_n+1)}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_n = 0}^{s_n-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots {s_n\\choose k_n}{s_n\\choose k_n+2}\\\\\\nonumber\n&\\sum_{{k'\\!\\!}_1=0}^{{s'\\!\\!}_1-2}\\ldots \\sum_{{k'\\!\\!}_n = 0}^{{s'\\!\\!}_n-2}{{s'\\!\\!}_1\\choose {k'\\!\\!}_1}{{s'\\!\\!}_1\\choose {k'\\!\\!}_1+2}\\ldots {{s'\\!\\!}_n\\choose {k'\\!\\!}_n}{{s'\\!\\!}_n\\choose {k'\\!\\!}_n+2}\\\\\\nonumber\n&\\frac{2^{n-1}}{n}\\sum_{\\sigma\\in P_n}\\sum_{\\rho\\in P_n}\n(s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)})!({s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)})!\\\\\\nonumber\n&\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)})!({s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)})!\\\\\\nonumber\n&\\frac{(x_{\\sigma(1)}-y_{\\rho(1)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}}}{\\left(\\rvert x_{\\sigma(1)}-y_{\\rho(1)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}+1}}\\frac{(y_{\\rho(1)}-x_{\\sigma(2)})_+^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}}}{\\left(\\rvert y_{\\rho(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}+1}}\\\\\n&\\ldots\\frac{(x_{\\sigma(n)}-y_{\\rho(n)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}}}{\\left(\\rvert x_{\\sigma(n)}-y_{\\rho(n)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}+1}}\n\\frac{(y_{\\rho(n)}-x_{\\sigma(1)})_+^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}}}{\\left(\\rvert y_{\\rho(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\n\n\n\\subsubsection{Extended basis}\n\nWe normalize our operators in such a way that the $2$-point correlators in the extended basis are equal for even $s$:\n\\begin{align}\n&\\langle {\\mathbb{A}}_{s_1}(x) {\\mathbb{A}}_{s_2}(y)\\rangle = \\langle \\bar{\\mathbb{B}}_{s_1}(x)\\bar{\\mathbb{B}} _{s_2}(y)\\rangle\n =  \\mathcal{A}_{s_1}(x,y) \\delta_{s_1 s_2} \n\\end{align}\nand for odd $s$:\n\\begin{align}\n\\langle \\tilde{\\mathbb{A}}_{s_1}(x)\\tilde{\\mathbb{A}}_{s_2}(y)\\rangle  =  \\mathcal{A}_{s_1}(x,y) \\delta_{s_1 s_2}\n\\end{align}\nwith:\n\\begin{align}\n\\nonumber\n\\mathcal{A}_{s}(x,y) =& \\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} 2^{2s}i^{2s}\n\\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s}\\sum_{k_2 = 0}^{s}{s\\choose k_1}{s\\choose k_1}{s\\choose k_2}{s\\choose k_2}(-1)^{s-k_2+k_1}\n(s-k_1+k_2)!(s+k_1-k_2)!\\\\\\nonumber\n=&\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} 2^{2s}i^{2s}\n (2s)! \\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\\\\\n&\\sum_{k_1 = 0}^{s}\\sum_{k_2 = 0}^{s}{s\\choose k_1}{s\\choose k_1}{s\\choose k_2}{s\\choose k_2}(-1)^{k_2+k_1}\\frac{1}{{2s\\choose k_1+k_2}}\n\\end{align}\nThe very same correlators are evaluated by a trick \\cite{Kazakov:2012ar} (appendix \\ref{appA3}):\n\\begin{align}\n \\mathcal{A}_{s}(x,y)\n= &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} 2^{2s} (-1)^s (2s)!\n\\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\n\\end{align}\nTherefore, we have discovered the following -- seemingly nontrivial -- identity (section \\ref{coordinates}):\n\\begin{align}\n\\delta_{s_1s_2}= \n\\sum_{k_1 = 0}^{s_1}\\sum_{k_2 = 0}^{s_2}{s_1\\choose k_1}{s_1\\choose k_1}{s_2\\choose k_2}{s_2\\choose k_2}(-1)^{k_2+k_1}\\frac{1}{{s_1+s_2\\choose k_1+k_2}}\n\\end{align}\nWe have not found a direct proof of the above identity, but we have verified it numerically. \\par\nMoreover, the only nonvanishing $3$-point correlators are:\n\\begin{align}\n\\label{defc3}\n\\langle {\\mathbb{A}}_{s_1}(x){\\mathbb{A}}_{s_2}(y){\\mathbb{A}}_{s_3}(z)\\rangle=\\langle {\\mathbb{A}}_{s_1}(x){\\mathbb{B}}_{s_2}(y)\\bar{\\mathbb{B}}_{s_3}(z)\\rangle= \\mathcal{A}_{s_1s_2s_3}(x,y,z)\n\\end{align}\nand:\n\\begin{align}\n\\label{defc3}\n\\langle {\\mathbb{A}}_{s_1}(x)\\tilde{\\mathbb{A}}_{s_2}(y)\\tilde{\\mathbb{A}}_{s_3}(z)\\rangle  = \\mathcal{A}_{s_1s_2s_3}(x,y,z)\n\\end{align}\nwith:\n\\begin{align}\n\\nonumber\n \\mathcal{A}_{s_1s_2s_3}(x,y,z)\n =&-\\frac{1}{(4\\pi^2)^3} (1+(-1)^{s_1+s_2+s_3})  \\frac{N^2-1}{8}i^{s_1+s_2+s_3}2^{s_1+s_2+s_3}\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1}\\sum_{k_2 = 0}^{s_2}\\sum_{k_3 = 0}^{s_3}{s_1\\choose k_1}{s_1\\choose k_1}{s_2\\choose k_2}{s_2\\choose k_2}{s_3\\choose k_3}{s_3\\choose k_3}\\\\\\nonumber\n&(s_1-k_1+k_2)!(s_2-k_2+k_3)!(s_3-k_3+k_1)!  \\\\\\nonumber\n&\\frac{(x-y)^{s_1-k_1+k_2}_+}{(\\rvert x-y\\rvert^2)^{s_1+1-k_1+k_2}}\\frac{(y-z)^{s_2-k_2+k_3}_+}{(\\rvert y-z\\rvert^2)^{s_2+1-k_2+k_3}}\\frac{(z-x)^{s_3-k_3+k_1}_+}{(\\rvert z-x\\rvert^2)^{s_3+1-k_3+k_1}}\\nonumber \\\\\n\\end{align}\nWe also compute the $n$-point correlators. In the balanced sector, we get:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{A}_{s_1}(x_1)\\ldots \\mathbb{A}_{s_n}(x_n)\\rangle_{conn} =\\frac{1}{(4\\pi^2)^n}\\frac{N^2-1}{2^n}2^{\\sum_{l=1}^n s_l}i^{\\sum_{l=1}^n s_l}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_n = 0}^{s_n}{s_1\\choose k_1}^2\\ldots {s_n\\choose k_n}^2\\frac{(-1)^n}{n}\\sum_{\\sigma\\in P_n}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n)}-x_{\\sigma(1)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\nThe very same formula holds for an even number of operators $\\tilde{\\mathbb{A}}_s$, otherwise the correlators vanish. \nThe nonvanishing correlators in the balanced sector are:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{A}_{s_1}(x_1)\\ldots \\mathbb{A}_{s_n}(x_n)\\tilde{\\mathbb{A}}_{s_{n+1}}(x_{n+1})\\ldots \\tilde{\\mathbb{A}}_{s_{n+2m}}(x_{n+2m})\\rangle_{conn} \\\\\\nonumber\n&=\\frac{1}{(4\\pi^2)^{n+2m}}\\frac{N^2-1}{2^{n+2m}}2^{\\sum_{l=1}^{n+2m} s_l}i^{\\sum_{l=1}^{n+2m} s_l}\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_{n+2m} = 0}^{s_{n+2m}}{s_1\\choose k_1}^2\\ldots{s_{n+2m}\\choose k_{n+2m}}^2\\\\\\nonumber\n&\\frac{(-1)^{n+2m}}{n+{2m}}\\sum_{\\sigma\\in P_{n+2m}}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n+2m)}-x_{\\sigma(1)})_+^{s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n+{2m})}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)}+1}}\n\\end{align}\nIn the unbalanced sector, we get:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{B}_{s_1}(x_1)\\ldots \\mathbb{B}_{s_n}(x_n)\\bar{\\mathbb{B}}_{{s'\\!\\!}_1}(y_1)\\ldots \\bar{\\mathbb{B}}_{{s'\\!\\!}_n}(y_n)\\rangle=\\frac{1}{(4\\pi^2)^{2n}}\\frac{N^2-1}{2^{2n}}2^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}i^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_n = 0}^{s_n}\\sum_{{k'\\!\\!}_1=0}^{{s'\\!\\!}_1-2}\\ldots \\sum_{{k'\\!\\!}_n = 0}^{{s'\\!\\!}_n}{s_1\\choose k_1}^2\\ldots {s_n\\choose k_n}^2{{s'\\!\\!}_1\\choose {k'\\!\\!}_1}^2\\ldots {{s'\\!\\!}_n\\choose {k'\\!\\!}_n}^2\\\\\\nonumber\n&\\frac{2^{n-1}}{n}\\sum_{\\sigma\\in P_n}\\sum_{\\rho\\in P_n}\n(s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)})!({s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)})!\\\\\\nonumber\n&\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)})!({s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)})!\\\\\\nonumber\n&\\frac{(x_{\\sigma(1)}-y_{\\rho(1)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}}}{\\left(\\rvert x_{\\sigma(1)}-y_{\\rho(1)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}+1}}\\frac{(y_{\\rho(1)}-x_{\\sigma(2)})_+^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}}}{\\left(\\rvert y_{\\rho(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}+1}}\\\\\n&\\ldots\\frac{(x_{\\sigma(n)}-y_{\\rho(n)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}}}{\\left(\\rvert x_{\\sigma(n)}-y_{\\rho(n)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}+1}}\n\\frac{(y_{\\rho(n)}-x_{\\sigma(1)})_+^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}}}{\\left(\\rvert y_{\\rho(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\n\n\n\n\n \\subsection{Euclidean $n$-point correlators }\n \n \\subsubsection{Standard basis}\n\n After the Wick rotation (appendix \\ref{appN} and section \\ref{8}), we obtain in the standard basis:\n \\begin{align}\n \\nonumber\n \\mathcal{C}^E_{s}(x,y)\n = &\\frac{1}{(4\\pi^2)^2}\\frac{N^2-1}{4} \\frac{2^{2s+2}}{(4!)^2}(s+1)^2(s+2)^2\n \\frac{(x-y)_{z}^{2s}}{((x-y)^2)^{2s+2}}\\\\\\nonumber\n &\\sum_{k_1 = 0}^{s-2}\\sum_{k_2 = 0}^{s-2}{s\\choose k_1}{s\\choose k_1+2}{s\\choose k_2+2}{s\\choose k_2}(-1)^{s-k_2+k_1}\\\\\n &(s-k_1+k_2)!(s+k_1-k_2)!\n  \\end{align}\nwhich is equivalent to:\n \\begin{align}\n \\mathcal{C}^E_{s}(x,y) = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{2s+2}}{(4!)^2}(s-1)s(s+1)(s+2)(2s)!\n \\frac{(x-y)_{z}^{2s}}{(( x-y)^2)^{2s+2}}\n \\end{align}\n For the nonvanishing $3$-point correlators, we get:\n \\begin{align}\n\\nonumber\n \\mathcal{C}^E_{s_1s_2s_3}(x,y,z)=&\\frac{1}{(4\\pi^2)^3} (-1)^{s_1+s_2+s_3}(1+(-1)^{s_1+s_2+s_3})   \\left(\\frac{2}{4!}\\right)^3\\frac{N^2-1}{8}2^{s_1+s_2+s_3}\\\\\\nonumber\n & (s_1+1)(s_1+2)(s_2+1)(s_2+2)(s_3+1)(s_3+2)\\\\\\nonumber\n &\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}\\sum_{k_3 = 0}^{s_3-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}{s_3\\choose k_3}{s_3\\choose k_3+2}\\\\\\nonumber\n &(s_1-k_1+k_2)!(s_2-k_2+k_3)!(s_3-k_3+k_1)!  \\\\\n &\\frac{(x-y)^{s_1-k_1+k_2}_{z}}{(( x-y)^2)^{s_1+1-k_1+k_2}}\\frac{(y-z)^{s_2-k_2+k_3}_{z}}{(( y-z)^2)^{s_2+1-k_2+k_3}}\\frac{(z-x)^{s_3-k_3+k_1}_{z}}{(( z-x)^2)^{s_3+1-k_3+k_1}}\n \\end{align}\nMoreover, for the $n$-point correlators in the balanced sector, we obtain:\n \\begin{align}\n \\nonumber\n &\\langle \\mathbb{O}^E_{s_1}(x_1)\\ldots \\mathbb{O}^E_{s_n}(x_n)\\rangle_{conn} =\\frac{1}{(4\\pi^2)^n}\\frac{N^2-1}{2^n}2^{\\sum_{l=1}^n s_l}(-1)^{\\sum_{l=1}^ns_l}\\\\\\nonumber\n &\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma(s_n+3)}{\\Gamma(5)\\Gamma(s_n+1)}\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_n = 0}^{s_n-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots {s_n\\choose k_n}{s_n\\choose k_n+2}\\\\\\nonumber\n &\\frac{1}{n}\\sum_{\\sigma\\in P_n}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)})!\\\\\n &\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_{z}^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(( x_{\\sigma(1)}-x_{\\sigma(2)})^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n)}-x_{\\sigma(1)})_{z}^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}}}{\\left(( x_{\\sigma(n)}-x_{\\sigma(1)})^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}+1}}\n \\end{align}\nThe very same formula holds for an even number of operators $\\tilde{\\mathbb{O}}^E_s$, otherwise the correlators vanish. The nonvanishing correlators in the balanced sector are:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{O}^E_{s_1}(x_1)\\ldots \\mathbb{O}^E_{s_n}(x_n)\\tilde{\\mathbb{O}}^E_{s_{n+1}}(x_{n+1})\\ldots \\tilde{\\mathbb{O}}^E_{s_{n+2m}}(x_{n+2m})\\rangle_{conn}\\\\\\nonumber\n&=\\frac{1}{(4\\pi^2)^{n+2m}}\\frac{N^2-1}{2^{n+2m}}2^{\\sum_{l=1}^{n+2m} s_l}(-1)^{\\sum_{l=1}^{n+2m}s_l}\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)}\\ldots\\frac{\\Gamma(3)\\Gamma(s_{n+2m}+3)}{\\Gamma(5)\\Gamma(s_{n+2m}+1)}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_{n+2m} = 0}^{s_{n+2m}-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots{s_{n+2m}\\choose k_{n+2m}}{s_{n+2m}\\choose k_{n+2m}+2}\\\\\\nonumber\n&\\frac{1}{n+{2m}}\\sum_{\\sigma\\in P_{n+2m}}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_{z}^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(( x_{\\sigma(1)}-x_{\\sigma(2)})^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n+2m)}-x_{\\sigma(1)})_{z}^{s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)}}}{\\left(( x_{\\sigma(n+{2m})}-x_{\\sigma(1)})^2\\right)^{s_{\\sigma(n+{2m})}-k_{\\sigma(n+{2m})}+k_{\\sigma(1)}+1}}\\,\n\\end{align}\nIn the unbalanced sector, we get:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{S}^E_{s_1}(x_1)\\ldots \\mathbb{S}^E_{s_n}(x_n)\\bar{\\mathbb{S}}^E_{{s'\\!\\!}_1}(y_1)\\ldots \\bar{\\mathbb{S}}^E_{{s'\\!\\!}_n}(y_n)\\rangle=\\frac{1}{(4\\pi^2)^{2n}}\\frac{N^2-1}{2^{2n}}2^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}(-1)^{\\sum_{l=1}^ns_l+{s'\\!\\!}_l}\\\\\\nonumber\n&\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma(s_n+3)}{\\Gamma(5)\\Gamma(s_n+1)}\\frac{\\Gamma(3)\\Gamma({s'\\!\\!}_1+3)}{\\Gamma(5)\\Gamma({s'\\!\\!}_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma({s'\\!\\!}_n+3)}{\\Gamma(5)\\Gamma({s'\\!\\!}_n+1)}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_n = 0}^{s_n-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots {s_n\\choose k_n}{s_n\\choose k_n+2}\\\\\\nonumber\n&\\sum_{{k'\\!\\!}_1=0}^{{s'\\!\\!}_1-2}\\ldots \\sum_{{k'\\!\\!}_n = 0}^{{s'\\!\\!}_n-2}{{s'\\!\\!}_1\\choose {k'\\!\\!}_1}{{s'\\!\\!}_1\\choose {k'\\!\\!}_1+2}\\ldots {{s'\\!\\!}_n\\choose {k'\\!\\!}_n}{{s'\\!\\!}_n\\choose {k'\\!\\!}_n+2}\\\\\\nonumber\n&\\frac{2^{n-1}}{n}\\sum_{\\sigma\\in P_n}\\sum_{\\rho\\in P_n}\n(s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)})!({s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)})!\\\\\\nonumber\n&\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)})!({s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)})!\\\\\\nonumber\n&\\frac{(x_{\\sigma(1)}-y_{\\rho(1)})_{z}^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}}}{\\left(( x_{\\sigma(1)}-y_{\\rho(1)})^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}+1}}\\frac{(y_{\\rho(1)}-x_{\\sigma(2)})_{z}^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}}}{\\left(( y_{\\rho(1)}-x_{\\sigma(2)})^2\\right)^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}+1}}\\\\\n&\\ldots\\frac{(x_{\\sigma(n)}-y_{\\rho(n)})_{z}^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}}}{\\left(( x_{\\sigma(n)}-y_{\\rho(n)})^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}+1}}\n\\frac{(y_{\\rho(n)}-x_{\\sigma(1)})_{z}^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}}}{\\left(( y_{\\rho(n)}-x_{\\sigma(1)})^2\\right)^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\n \n \\subsubsection{Extended basis}\n \n We obtain in the extended basis:\n  \\begin{align}\n \\nonumber\n\\mathcal{A}^E_{s}(x,y)\n  = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} 2^{2s}\n \\frac{(x-y)_{z}^{2s}}{(( x-y)^2)^{2s+2}}\\\\\n &\\sum_{k_1 = 0}^{s}\\sum_{k_2 = 0}^{s}{s\\choose k_1}{s\\choose k_1}{s\\choose k_2}{s\\choose k_2}(-1)^{s-k_2+k_1}\n (s-k_1+k_2)!(s+k_1-k_2)!\n \\end{align}\n which is equivalent to:\n \\begin{align}\n \\mathcal{A}^E_{s}(x,y)=&\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} 2^{2s} (2s)! \n \\frac{x_{{z}}^{2s}}{(x^2)^{2s+2}}\n \\end{align}\n For the nonvanishing $3$-point correlators, we get:\n \\begin{align}\n\\nonumber\n \\mathcal{A}^E_{s_1s_2s_3}(x,y,z) =&\\frac{1}{(4\\pi^2)^3} (-1)^{s_1+s_2+s_3}(1+(-1)^{s_1+s_2+s_3})   \\frac{N^2-1}{8}2^{s_1+s_2+s_3}\\\\\\nonumber\n &\\sum_{k_1 = 0}^{s_1}\\sum_{k_2 = 0}^{s_2}\\sum_{k_3 = 0}^{s_3}{s_1\\choose k_1}{s_1\\choose k_1}{s_2\\choose k_2}{s_2\\choose k_2}{s_3\\choose k_3}{s_3\\choose k_3}\\\\\\nonumber\n &(s_1-k_1+k_2)!(s_2-k_2+k_3)!(s_3-k_3+k_1)!  \\\\\\nonumber\n &\\frac{(x-y)^{s_1-k_1+k_2}_{z}}{(( x-y)^2)^{s_1+1-k_1+k_2}}\\frac{(y-z)^{s_2-k_2+k_3}_{z}}{(( y-z)^2)^{s_2+1-k_2+k_3}}\\frac{(z-x)^{s_3-k_3+k_1}_{z}}{(( z-x)^2)^{s_3+1-k_3+k_1}} \\\\\n\\end{align}\n Moreover, for the $n$-point correlators in the balanced sector, we obtain:\n \\begin{align}\n \\nonumber\n&\\langle \\mathbb{A}^E_{s_1}(x_1)\\ldots \\mathbb{A}^E_{s_n}(x_n)\\rangle_{conn} =\\frac{1}{(4\\pi^2)^n}\\frac{N^2-1}{2^n}2^{\\sum_{l=1}^n s_l}(-1)^{\\sum_{l=1}^ns_l}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_n = 0}^{s_n}{s_1\\choose k_1}^2\\ldots {s_n\\choose k_n}^2\\frac{1}{n}\\sum_{\\sigma\\in P_n}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)})!\\\\\n &\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_{{z}}^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left((  x_{\\sigma(1)}-x_{\\sigma(2)})^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n)}-x_{\\sigma(1)})_{{z}}^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}}}{\\left(( x_{\\sigma(n)}-x_{\\sigma(1)})^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}+1}}\n \\end{align}\n The very same formula holds for an even number of $\\tilde{\\mathbb{A}}^E_s$ operators, otherwise the correlators vanish. The nonvanishing correlators in the balanced sector are:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{A}^E_{s_1}(x_1)\\ldots \\mathbb{A}^E_{s_n}(x_n)\\tilde{\\mathbb{A}}^E_{s_{n+1}}(x_{n+1})\\ldots \\tilde{\\mathbb{A}}^E_{s_{n+2m}}(x_{n+2m})\\rangle_{conn} \\\\\\nonumber\n&=\\frac{1}{(4\\pi^2)^{n+2m}}\\frac{N^2-1}{2^{n+2m}}2^{\\sum_{l=1}^{n+2m} s_l}(-1)^{\\sum_{l=1}^{n+2m}s_l}\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_{n+2m} = 0}^{s_{n+2m}}{s_1\\choose k_1}^2\\ldots{s_{n+2m}\\choose k_{n+2m}}^2\\\\\\nonumber\n&\\frac{1}{n+2m}\\sum_{\\sigma\\in P_{n+2m}}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_{z}^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(( x_{\\sigma(1)}-x_{\\sigma(2)})^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n+2m)}-x_{\\sigma(1)})_{z}^{s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)}}}{\\left(( x_{\\sigma(n+2m)}-x_{\\sigma(1)})^2\\right)^{s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)}+1}}\n\\end{align} \nIn the unbalanced sector, we get:\n \\begin{align}\n \\nonumber\n &\\langle \\mathbb{B}^E_{s_1}(x_1)\\ldots \\mathbb{B}^E_{s_n}(x_n)\\bar{\\mathbb{B}}^E_{{s'\\!\\!}_1}(y_1)\\ldots \\bar{\\mathbb{B}}^E_{{s'\\!\\!}_n}(y_n)\\rangle=\\frac{1}{(4\\pi^2)^{2n}}\\frac{N^2-1}{2^{2n}}2^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}(-1)^{\\sum_{l=1}^ns_l+{s'\\!\\!}_l}\\\\\\nonumber\n &\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_n = 0}^{s_n}\\sum_{{k'\\!\\!}_1=0}^{{s'\\!\\!}_1-2}\\ldots \\sum_{{k'\\!\\!}_n = 0}^{{s'\\!\\!}_n}{s_1\\choose k_1}^2\\ldots {s_n\\choose k_n}^2{{s'\\!\\!}_1\\choose {k'\\!\\!}_1}^2\\ldots {{s'\\!\\!}_n\\choose {k'\\!\\!}_n}^2\\\\\\nonumber\n&\\frac{2^{n-1}}{n}\\sum_{\\sigma\\in P_n}\\sum_{\\rho\\in P_n}\n(s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)})!({s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)})!\\\\\\nonumber\n&\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)})!({s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)})!\\\\\\nonumber\n&\\frac{(x_{\\sigma(1)}-y_{\\rho(1)})_{z}^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}}}{\\left(( x_{\\sigma(1)}-y_{\\rho(1)})^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}+1}}\\frac{(y_{\\rho(1)}-x_{\\sigma(2)})_{z}^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}}}{\\left(( y_{\\rho(1)}-x_{\\sigma(2)})^2\\right)^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}+1}}\\\\\n&\\ldots\\frac{(x_{\\sigma(n)}-y_{\\rho(n)})_{z}^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}}}{\\left(( x_{\\sigma(n)}-y_{\\rho(n)})^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}+1}}\n\\frac{(y_{\\rho(n)}-x_{\\sigma(1)})_{z}^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}}}{\\left(( y_{\\rho(n)}-x_{\\sigma(1)})^2\\right)^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}+1}}\n \\end{align}\n \n \n \\section{Plan of the paper}\n \nIn section \\ref{0} we outline our main results and physics motivations. \\par\nIn section \\ref{00} we display our results for the correlators to the lowest perturbative order both in Minkowskian and Euclidean space-time. \\par\nIn section \\ref{operators} we review the classification and construction of the gluonic operators with collinear twist $2$ in Minkowskian space-time both in the standard and extended basis.\\par\nIn section \\ref{coordinates} we compute the $2$-point correlators in Minkowskian space-time both in the standard and extended basis.\\par\nIn section \\ref{threepoint} we compute the $3$-point correlators in Minkowskian space-time both in the standard and extended basis.\\par\nIn section \\ref{npoint} we compute the $n$-point correlators in Minkowskian space-time both in the standard and extended basis in the balanced and unbalanced sectors separately. \\par\nIn section \\ref{8} we compute the $n$-point correlators in Euclidean space-time either by analytic continuation or by employing the corresponding Euclidean operators.\\par\nIn the appendices we fix the notation and provide ancillary computations.\\par\nIn appendix \\ref{appcheck} we verify that our results for the $2$- and $3$-point correlators of balanced operators with even collinear spin in Minkowskian space-time coincide with \\cite{Kazakov:2012ar}. \n\n\\section{Twist-$2$ gluonic operators in Minkowskian space-time \\label{operators}}\n\n\nWe review the construction of the standard and extended conformal bases for gauge-invariant gluonic operators with $\\tau=2$. We also work out the dictionary between the spinorial, vectorial and complex bases (appendices \\ref{appA2} and \\ref{appC}).\n\n\n\n\\subsection{Standard basis \\label{standardb}}\n\n\nTo construct gauge-invariant gluonic operators with $\\tau=2$ that are primary (section \\ref{00}) for the collinear conformal group, we should find -- according to eq. (\\ref{maximum2}) -- suitable gauge-covariant elementary conformal operators. \\par\nThe local gauge-covariant operator with lowest canonical dimension, $d=2$, in YM theory is the field-strength tensor, $F_{\\mu\\nu}=\\partial_{\\mu}A_\\nu-\\partial_{\\nu}A_\\nu+ig\\left[A_\\mu,A_\\nu\\right]$, where $A_{ \\mu}=A_\\mu^a T^a$ is a traceless Hermitian matrix, with $T^{a}$ the generators of the $SU(N)$ Lie algebra:\n\\begin{equation}\n[T^a,T^b]=if^{abc}T^c\n\\end{equation}\nnormalized in the standard way:\n\\begin{equation}\n\\label{colour_trace}\n\\Tr (T^aT^b)=\\frac{1}{2}\\delta^{ab}\n\\end{equation}\nIt is convenient to write $F_{\\mu\\nu}$ in the spinorial representation \\cite{Braun:2008ia} (appendix \\ref{appA2}):\n\\begin{equation}\nF_{a\\dot{a}b\\dot{b}} = \\sigma_{a\\dot{a}}^\\mu\\sigma_{b\\dot{b}}^\\nu F_{\\mu\\nu}\n\\end{equation}\nIt turns out \\cite{Braun:2008ia} that:\n\\begin{equation}\nF_{a\\dot{a}b\\dot{b}} = 2\\left(f_{ab}\\epsilon_{\\dot{a}\\dot{b}}- \\epsilon_{ab} f_{\\dot{a}\\dot{b}}\\right)\n\\end{equation}\ndecomposes into the sum of two chiral representations, $f_{ab} \\in (1,0)$ and $f_{\\dot{a}\\dot{b}}\\in (0,1)$, of spin $S=1$ (appendix \\ref{appA2}). \nIn Minkowskian space-time $f_{\\dot{a}\\dot{b}} = \\bar{f}_{ab}$.\n$f_{11}$ and $f_{\\dot{1}\\dot{1}}$ have maximal collinear spin (appendix \\ref{appA2}), $s = 1$, along the $p_+$ direction. Therefore, they have $\\tau = d-s = 1$ and $j = s+\\frac{\\tau}{2} = \\frac{3}{2}$.\nHence, they are well suited (section \\ref{00}) to build $2$-gluon twist-$2$ primary conformal operators.\nTaking the tensor product of the above representations, we get:\n\\begin{align}\n&(1,0)\\otimes(0,1) = (1,1)_+\\oplus(1,1)_-\\\\\\nonumber\n&(1,0)\\otimes(1,0) = (2,0)\\oplus\\ldots\\\\\\nonumber\n&(0,1)\\otimes(0,1) = (0,2)\\oplus\\ldots\n\\end{align}\nwhere $+$ and $-$ label the parity, and the dots denote terms that do not contribute to the components with maximal collinear spin. Hence, there are four operators with maximal $s$ that can be constructed by means of the corresponding bilinear operators:\n\\begin{align} \\label{44}\n&O(x_1,x_2) = f_{11}(x_1) f_{\\dot{1}\\dot{1}}(x_2) + f_{11}(x_2) f_{\\dot{1}\\dot{1}}(x_1)\\\\\\nonumber\n&O(x_1,x_2) = f_{11}(x_1) f_{\\dot{1}\\dot{1}}(x_2)- f_{11}(x_2) f_{\\dot{1}\\dot{1}}(x_1)\\\\\\nonumber\n&S(x_1,x_2)= \\frac{1}{\\sqrt{2}}f_{11}(x_1)f_{11}(x_2)\\\\\\nonumber\n&\\bar{S}(x_1,x_2)= \\frac{1}{\\sqrt{2}}f_{\\dot{1}\\dot{1}}(x_2) f_{\\dot{1}\\dot{1}}(x_1)\n\\end{align}\nFollowing \\cite{Ohrndorf:1981qv, Braun:2003rp}, we build (section \\ref{00}) conformal operators with $\\tau=2$ and higher collinear spin inserting in eq. \\eqref{44} the Gegenbauer polynomials (appendix \\ref{appB}), $C^{\\alpha}_l(v)$, in the derivatives and afterwards taking the local limit $x_1=x_2$. $l$ is the order of the polynomial, and its relation to the collinear conformal spin is $j =l+j_1+j_2$, where $j_1$ and $j_2$ are the collinear conformal spins of the elementary operators, and $\\alpha = 2j_1-\\frac{1}{2}$. The Gegenbauer polynomials are either symmetric or antisymmetric for the substitution $v \\rightarrow-v$ for $l$ even or odd respectively (appendix \\ref{appB}). \nThe corresponding primary conformal operators match precisely the ones in \\cite{Belitsky:2003sh, Beisert:2004fv} up to perhaps the overall normalization:\n\\begin{align}\n\\label{basis}\n\\nonumber\n&\\mathbb{O}_{s} = \\Tr f_{11}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right) f_{\\dot{1}\\dot{1}}(x) \\qquad s = 2,4,6,\\ldots\\\\\\nonumber\n&\\tilde{\\mathbb{O}}_{s} = \\Tr f_{11}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right) f_{\\dot{1}\\dot{1}}(x) \\qquad s = 3,5,7,\\ldots\\\\\\nonumber\n&\\mathbb{S}_{s} =\\frac{1}{\\sqrt{2}}\\Tr f_{11}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)f_{11}(x) \\qquad s = 2,4,6,\\ldots\\\\\n&\\bar{\\mathbb{S}}_{s} =\\frac{1}{\\sqrt{2}}\\Tr {f}_{\\dot{1}\\dot{1}}(x)(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right){f}_{\\dot{1}\\dot{1}}(x)\\qquad s = 2,4,6,\\ldots\n\\end{align}\nwith $j = s+1$, $l =s-2$ and $\\alpha = \\frac{5}{2}$.\nFor brevity, we define as in \\cite{Kazakov:2012ar}:\n\\begin{equation}\n\\label{eq46}\n\\mathcal{G}_l^\\alpha(D_{x^+_1},D_{x^+_2}) = i^l (\\overrightarrow{D}_{x^+_2}+\\overleftarrow{D}_{x^+_1})^l C^{\\alpha}_l\\left(\\frac{\\overrightarrow{D}_{x^+_2}-\\overleftarrow{D}_{x^+_1}}{\\overrightarrow{D}_{x^+_2}+\\overleftarrow{D}_{x^+_1}}\\right)\n\\end{equation}\nFor $l =s-2$ and $\\alpha = \\frac{5}{2}$, we get in the light-cone gauge by means eq. (\\ref{physicalgegen}):\n\\begin{align}\n\\label{defg52}\n\\mathcal{G}^{\\frac{5}{2}}_{s-2}(\\partial_{x^+_1},\\partial_{x^+_2}) &=i^{s-2}\\frac{\\Gamma(s+3)\\Gamma(3)}{\\Gamma(5)\\Gamma(s+1)}\\sum_{k=0}^{s-2} {s\\choose k}{s\\choose k+2}(-1)^{s-k} \\overleftarrow{\\partial}_{x^+_1}^{s-k-2} \\overrightarrow{\\partial}_{x^+_2}^k\\nonumber\\\\\n &=\\frac{2i^{s-2}(s+1)(s+2)}{4!}\\sum_{k=0}^{s-2} {s\\choose k}{s\\choose k+2}(-1)^{s-k} \\overleftarrow{\\partial}_{x^+_1}^{s-k-2} \\overrightarrow{\\partial}_{x^+_2}^k\n\\end{align}\n\n\n\\subsection{Extended basis \\label{nonstandardb}}\n\n\n\nThere exists another choice of the basis for primary conformal operators with $\\tau=2$ involving the elementary operators $D_{+}^{-1}f_{11}$ and $D_{+}^{-1} f_{\\dot{1}\\dot{1}}$, which are nonlocal in general, but local in the light-cone gauge. Yet, gauge invariance ensures that the corresponding gauge-invariant correlators are local. Indeed, in the light-cone gauge (appendix \\ref{appC}):\n\\begin{align}\n\\partial_{+}^{-1}f_{11} = -\\bar{A}\\nonumber \\\\\n \\partial_{+}^{-1} f_{\\dot{1}\\dot{1}} =  -{A} \n\\end{align}\nwhere $A$ has $d = 1$, $s = 0$, $ j =\\frac{1}{2}$ and $\\tau = 1$. The corresponding operators with $\\tau=2$ read:\n\\begin{align}\n\\nonumber\n\\label{basisSquasilocal}\n&\\mathbb{A}_{s} = \\Tr D_{+}^{-1}f_{11}(x)\\,(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)D_{+}^{-1} f_{\\dot{1}\\dot{1}}(x)  \\qquad s = 0,2,4,\\ldots\\\\\\nonumber\n&\\tilde{\\mathbb{A}}_{s} = \\Tr D_{+}^{-1}f_{11}(x)\\,(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right) D_{+}^{-1} f_{\\dot{1}\\dot{1}}(x)  \\qquad s = 1,3,5,\\ldots\\\\\\nonumber\n&\\mathbb{B}_{s} =\\frac{1}{\\sqrt{2}}\\Tr D_{+}^{-1}f_{11}(x)\\,(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)D_{+}^{-1}f_{11}(x)\\qquad s = 0,2,4,\\ldots\\\\\n&\\bar{\\mathbb{B}}_{s} =\\frac{1}{\\sqrt{2}}\\Tr D_{+}^{-1} f_{\\dot{1}\\dot{1}}(x)\\,(i\\overrightarrow{D}_++i\\overleftarrow{D}_+)^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_+-\\overleftarrow{D}_+}{\\overrightarrow{D}_++\\overleftarrow{D}_+}\\right)D_{+}^{-1} f_{\\dot{1}\\dot{1}}(x) \\qquad s = 0,2,4,\\ldots\n\\end{align}\nwith $j = s+1$, $l =s$ and $\\alpha = \\frac{1}{2}$. This basis naturally arises in SUSY calculations \\cite{Belitsky:2004sc}, and also includes (nonlocal) operators with $s=0,1$.\nWe obtain in the light-cone gauge by means of eq. (\\ref{physicalgegen2}):\n\\begin{equation} \\label{AA}\n\\mathcal{G}^{\\frac{1}{2}}_{s}(\\partial_{x^+_1},\\partial_{x^+_2}) = i^s\\sum_{k = 0}^{s}{s\\choose k}{s\\choose k}(-1)^{s-k}\\overleftarrow{\\partial}_{x^+_1}^{s-k}\\overrightarrow{\\partial}_{x^+_2}^k\n\\end{equation}\n\n\n\n\\section{$2$-point correlators of twist-$2$ gluonic operators\\label{coordinates}}\n\n\n\nWe compute to the lowest perturbative order the $2$-point correlators of the operators in both bases.\n\n\\subsection{Standard basis}\n\nIn the light-cone gauge, the $2$-point correlators of the balanced operators with even $s$ are given by:\n\\begin{align}\n\\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle =&  \\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+}) \\nonumber \\\\\n&\\langle\\Tr f_{11}(x_1) f_{\\dot{1}\\dot{1}}(x_2)\\,\\Tr f_{11}(y_1) f_{\\dot{1}\\dot{1}}(y_2)\\rangle\\Big\\rvert_{x_1=x_2=x}^{y_1=y_2=y}\n\\end{align}\nThere is only one Wick contraction:\n\\begin{align}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle =&  \\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\\\\n&\\langle\\Tr f_{11}(x_1) f_{\\dot{1}\\dot{1}}(y_2)\\rangle\\langle\\Tr f_{11}(y_1) f_{\\dot{1}\\dot{1}}(x_2)\\rangle\\Big\\rvert_{x_1=x_2=x}^{y_1=y_2=y}\n\\end{align}\nBy means of eq. \\eqref{axialprop2}, we get:\n\\begin{align}\n\\label{scalarcorr}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle =& \\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4}\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\\\\\nonumber\n&\\partial^+_{x_1}\\partial^+_{x_2}\\partial^+_{y_1}\\partial^+_{y_2}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-x_2\\rvert^2}\\Big\\rvert_{x_1=x_2=x}^{y_1=y_2=y}\\\\\n\\end{align}\nNow we substitute eq. (\\ref{defg52}) into the above equation:\n\\begin{align}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^2i^{s_1+s_2-4}}{(4!)^2}(s_1+1)(s_1+2)(s_2+1)(s_2+2)\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}\\\\\n&(-\\partial_{x_1^+})^{s_1-k_1-1}\\partial^{k_1+1}_{x_2^+}(-\\partial_{y_1^+})^{s_2-k_2-1}\\partial^{k_2+1}_{y_2^+}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-x_2\\rvert^2}\\Big\\rvert_{x_1=x_2=x}^{y_1=y_2=y}\n\\end{align}\nWe compute the derivatives:\n\\begin{align}\n\\nonumber\n\\label{doubleder}\n&\\partial_{x^+}^{i}\\partial_{y^+}^{j}\\frac{1}{\\rvert x-y\\rvert^2} =\\partial_{x^+}^{i}\\partial_{y^+}^{j}\\frac{1}{2 (x-y)_+(x-y)_--(x-y)^2_\\perp} \\\\\\nonumber\n&=(-1)^{j}\\partial_{x^+}^{i+j}\\frac{1}{2 (x-y)_+(x-y)_--(x-y)^2_\\perp} \\\\\n&=(-1)^{i} (i+j)!\\,2^{i+j} \\frac{(x-y)_+^{i+j}}{(\\rvert x-y\\rvert^2)^{i+j+1}}\n\\end{align}\nby induction on the index $i$:\n\\begin{align}\n\\nonumber\n&\\partial_{x^+}^{i+1}\\partial_{y^+}^{j}\\frac{1}{\\rvert x-y\\rvert^2} =(-1)^{i} (i+j)!\\,2^{i+j} \\partial_{x^+} \\frac{(x-y)_+^{i+j}}{(\\rvert x-y\\rvert^2)^{i+j+1}}\\\\\\nonumber\n&=(-1)^{i} (i+j)!\\,2^{i+j} \\partial_{x_-}\\frac{(x-y)_+^{i+j}}{(2 (x-y)_+(x-y)_--(x-y)^2_\\perp)^{i+j+1}}\\\\\\nonumber\n&=(-1)^{i} (i+j)!\\,2^{i+j} (x-y)_+^{i+j} \\frac{-2(i+j+1)(x-y)_+ }{(2 (x-y)_+(x-y)_--(x-y)^2_\\perp)^{i+j+2}}\\\\\n&=(-1)^{i+1} (i+j+1)!\\,2^{i+j+1} \\frac{(x-y)_+^{i+j+1}}{(\\rvert x-y\\rvert^2)^{i+j+2}}\n\\end{align}\nWe obtain:\n\\begin{align}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x) &\\mathbb{O}_{s_2}(y)\\rangle = \\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^2i^{s_1+s_2-4}}{(4!)^2}(s_1+1)(s_1+2)(s_2+1)(s_2+2)\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}\\\\\\nonumber\n&(-1)^{s_1-k_1-1}\\, (-1)^{s_1-k_1-1}2^{s_1-k_1+k_2}(s_1-k_1+k_2)!\\frac{(x_1-y_2)_+^{s_1-k_1+k_2}}{(\\rvert x_1-y_2\\rvert^2)^{s_1-k_1+k_2+1}}\\\\\n&(-1)^{s_2-k_2-1}\\,(-1)^{s_2-k_2-1}2^{s_2+k_1-k_2}(s_2+k_1-k_2)!\\frac{(y_1-x_2)_+^{s_2+k_1-k_2}}{(\\rvert y_1-x_2\\rvert^2)^{s_2+k_1-k_2+1}}\\Big\\rvert_{x_1=x_2=x}^{y_1=y_2=y}\n\\end{align}\nthat becomes:\n\\begin{align}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{s_1+s_2+2}i^{s_1+s_2-4}}{(4!)^2}(s_1+1)(s_1+2)(s_2+1)(s_2+2)\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}\\\\\n&(s_1-k_1+k_2)!(s_2+k_1-k_2)!\\frac{(x-y)_+^{s_1-k_1+k_2}}{(\\rvert x-y\\rvert^2)^{s_1-k_1+k_2+1}}\\frac{(y-x)_+^{s_2+k_1-k_2}}{(\\rvert y-x\\rvert^2)^{s_2+k_1-k_2+1}}\n\\end{align}\nand simplifies as follows:\n\\begin{align}\n\\label{59}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle =& \\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{s_1+s_2+2}i^{s_1+s_2-4}}{(4!)^2}\\\\\\nonumber\n&(s_1+1)(s_1+2)(s_2+1)(s_2+2)\\frac{(x-y)_+^{s_1+s_2}}{(\\rvert x-y\\rvert^2)^{s_1+s_2+2}}\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}\\\\\\nonumber\n&(-1)^{s_2-k_2+k_1}(s_1-k_1+k_2)!(s_2+k_1-k_2)!  \\nonumber \\\\\n=& \\mathcal{C}_{s_1}(x,y)\\delta_{s_1s_2}\n\\end{align}\nsince the correlator is zero for $s_1\\neq s_2$ (appendix \\ref{appA3}).\nBy setting $s=s_1=s_2$, we get:\n\\begin{align}  \\label{CC}\n\\nonumber\n \\mathcal{C}_{s}(x,y) = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{2s+2}i^{2s-4}}{(4!)^2}(s+1)^2(s+2)^2\n\\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s-2}\\sum_{k_2 = 0}^{s-2}{s\\choose k_1}{s\\choose k_1+2}{s\\choose k_2}{s\\choose k_2+2}(-1)^{s-k_2+k_1}\\\\\n&(s-k_1+k_2)!(s+k_1-k_2)! \n\\end{align}\nMoreover, by the substitution in eq. (\\ref{59}):\n\\begin{equation}\n{k'\\!\\!}_2=s_2-2-k_2\n\\end{equation}\nwe obtain:\n\\begin{align}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{s_1+s_2+2}i^{s_1+s_2-4}}{(4!)^2}\\\\\\nonumber\n&(s_1+1)(s_1+2)(s_2+1)(s_2+2)\n\\frac{(x-y)_+^{s_1+s_2}}{(\\rvert x-y\\rvert^2)^{s_1+s_2+2}}\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{{k'\\!\\!}_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose s_2-2-{k'\\!\\!}_2}{s_2\\choose s_2-{k'\\!\\!}_2}(-1)^{s_2-s_2+2+{k'\\!\\!}_2+k_1}\\\\\n&(s_1-k_1+s_2-2-{k'\\!\\!}_2)!(s_2+k_1-s_2+2+{k'\\!\\!}_2)!\n\\end{align}\nthat becomes:\n\\begin{align}\n\\label{qesum}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{s_1+s_2+2}i^{s_1+s_2-4}}{(4!)^2}\\\\\\nonumber\n&(s_1+1)(s_1+2)(s_2+1)(s_2+2)\n\\frac{(x-y)_+^{s_1+s_2}}{(\\rvert x-y\\rvert^2)^{s_1+s_2+2}}\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2+2}{s_2\\choose k_2}(-1)^{k_2+k_1}\\\\\\nonumber\n&(s_1+s_2-k_1-k_2-2)!(k_1+k_2+2)!\\\\\\nonumber\n=&\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{s_1+s_2+2}i^{s_1+s_2-4}}{(4!)^2}\\\\\\nonumber\n&(s_1+1)(s_1+2)(s_2+1)(s_2+2)\n(s_1+s_2)!\\frac{(x-y)_+^{s_1+s_2}}{(\\rvert x-y\\rvert^2)^{s_1+s_2+2}}\\\\\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2+2}{s_2\\choose k_2}(-1)^{k_2+k_1} \\frac{1}{{s_1+s_2\\choose k_1+k_2+2}}\n\\end{align}\nBesides, according to the trick in \\cite{Kazakov:2012ar} (appendix \\ref{appA3}), we get:\n \\begin{align} \\label{qesum1}\n \\langle \\mathbb{O}_{s_1}(x) \\mathbb{O}_{s_2}(y)\\rangle =& \\delta_{s_1s_2}\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} \\frac{2^{2{s_1}+2}}{(4!)^2}(-1)^{s_1}\\nonumber\\\\\n &({s_1}-1){s_1}({s_1}+1)({s_1}+2)(2{s_1})!\n \\frac{(x-y)_+^{2{s_1}}}{(\\rvert x-y\\rvert^2)^{2{s_1}+2}}\n \\end{align}\nHence, comparing eqs. \\eqref{qesum} and \\eqref{qesum1}, we can virtually perform the sums in eq. \\eqref{qesum} to obtain the identity: \n\\begin{align}\n\\label{id2}\n\\delta_{s_1s_2}\\frac{s_1(s_1-1)}{(s_1+1)(s_1+2)}= &\n\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}(-1)^{k_2+k_1}\\frac{1}{{s_1+s_2\\choose k_1+k_2+2}}\n \\end{align}\nSimilarly, for the balanced operators with odd $s$, we get as well:\n\\begin{equation}\n\\langle \\tilde{\\mathbb{O}}_{s_1}(x) \\tilde{\\mathbb{O}}_{s_2}(y)\\rangle =\\mathcal{C}_{s_1}(x,y) \\delta_{s_1s_2}\n\\end{equation}\nwhere the definition of $\\mathcal{C}_{s}(x,y)$ in eq. \\eqref{CC} has been extended to odd $s$. Correspondingly, eq. \\eqref{id2} extends to odd $s_1,s_2$ as well. \\par\nNow we compute the only correlators of two unbalanced operators that are nonzero:\n\\begin{align}\n\\nonumber\n\\langle\\mathbb{S}_{s_1}(x)\\bar{\\mathbb{S}}_{s_2}(y)\\rangle =&\\frac{1}{2}\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\\\\n&\\langle\\Tr f_{11}(x_1)f_{11}(x_2)\\,\\Tr f_{\\dot{1}\\dot{1}}(y_1) f_{\\dot{1}\\dot{1}}(y_2)\\rangle\\Big\\rvert_{x_1=x_2=x}^{y_1=y_2=y}\n\\end{align}\nThere are two Wick contractions but an extra factor of $\\frac{1}{2}$ in the normalization of the operators, in such a way that the result is the same as for the correlators of balanced operators with even $s$:\n\\begin{align}\n\\langle\\mathbb{S}_{s_1}(x)\\bar{\\mathbb{S}}_{s_2}(y)\\rangle =& \\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4}\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\\\\\nonumber\n&\\partial^+_{x_1}\\partial^+_{x_2}\\partial^+_{y_1}\\partial^+_{y_2}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-x_2\\rvert^2}\\Big\\rvert_{x_1=x_2=x}^{y_1=y_2=y}\\\\\\nonumber\n=&\\mathcal{C}_{s_1}(x,y)\\delta_{s_1s_2}\n\\end{align}\nAll the remaining $2$-point correlators vanish.\n\n\n\\subsection{Extended basis}\n\n\nSimilarly, employing eq. \\eqref{AA}, we obtain the correlators in the extended basis:\n\\begin{align}\n\\label{semilocal}\n\\nonumber\n\\langle \\mathbb{A}_{s}(x) \\mathbb{A}_{s}(y)\\rangle = &\\frac{1}{(4\\pi^2)^2} \\frac{N^2-1}{4} 2^{2s}i^{2s}\n\\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\\\\\n&\\sum_{k_1 = 0}^{s}\\sum_{k_2 = 0}^{s}{s\\choose k_1}{s\\choose k_1}{s\\choose k_2}{s\\choose k_2}(-1)^{s-k_2+k_1}\n(s-k_1+k_2)!(s+k_1-k_2)! \\nonumber\\\\\n= & \\mathcal{A}_{s}(x,y)\n\\end{align}\nWe extend the above definition of $\\mathcal{A}_{s}(x,y)$ to odd $s$.\nWe get for even $s$:\n\\begin{align}\n\\langle\\mathbb{B}_{s_1}(x)\\bar{\\mathbb{B}}_{s_2}(y)\\rangle \n= &\\mathcal{A}_{s_1}(x,y)  \\delta_{s_1s_2} \n\\nonumber \\\\\n\\end{align}\nand for odd $s$:\n\\begin{align}\n\\langle \\tilde{\\mathbb{A}}_{s_1}(x) \\tilde{\\mathbb{A}}_{s_2}(y)\\rangle\n=&\\mathcal{A}_{s_1}(x,y) \\delta_{s_1s_2} \n\\end{align}\nWe obtain (appendix \\ref{appA3}):\n\\begin{align}\n\\mathcal{A}_{s}(x,y) = &\\frac{1}{(4\\pi^2)^2} (-1)^s\\frac{N^2-1}{4} 2^{2s}(2s)!\n\\frac{(x-y)_+^{2s}}{(\\rvert x-y\\rvert^2)^{2s+2}}\n\\end{align}\nSimilarly, it follows the identity:\n\\begin{align}\n\\label{id1}\n\\delta_{s_1s_2}= &\n\\sum_{k_1 = 0}^{s_1}\\sum_{k_2 = 0}^{s_2}{s_1\\choose k_1}{s_1\\choose k_1}{s_2\\choose k_2}{s_2\\choose k_2}(-1)^{k_2+k_1}\\frac{1}{{s_1+s_2\\choose k_1+k_2}}\n\\end{align}\nAll the remaining $2$-point correlators vanish.\n\n\n\n\\section{$3$-point correlators of twist-$2$ gluonic operators \\label{threepoint}}\n\n\nWe compute to the lowest perturbative order the $3$-point correlators of the operators in both bases.\n\n\n\\subsection{Standard basis} \n\n\nIn the light-cone gauge, for the $3$-point correlators of the balanced operators with even $s$, we obtain:\n\\begin{align}\n\\nonumber\n\\langle {\\mathbb{O}}_{s_1}(x){\\mathbb{O}}_{s_2}(y){\\mathbb{O}}_{s_3}&(z)\\rangle =\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\n&\\langle\\Tr f_{11}(x_1) f_{\\dot{1}\\dot{1}}(x_2)\\,\\Tr f_{11}(y_1) f_{\\dot{1}\\dot{1}}(y_2)\\,\\Tr f_{11}(z_1) f_{\\dot{1}\\dot{1}}(z_2)\\rangle\\Big\\rvert_{x_1=x_2=x}^{y_1=y_2=y,z_1=z_2=z}\n\\end{align}\nSince there are two Wick contractions, employing eq. \\eqref{axialprop2}, we get:\n\\begin{align}\n\\label{62}\n\\nonumber\n\\langle {\\mathbb{O}}_{s_1}(x)&{\\mathbb{O}}_{s_2}(y){\\mathbb{O}}_{s_3}(z)\\rangle =\\frac{1}{(4\\pi^2)^3}\\frac{N^2-1}{8}\\Bigg(\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\\nonumber\n&(-\\partial_{x_1^+})\\partial_{x_2^+}(-\\partial_{y_1^+})\\partial_{y_2^+}(-\\partial_{z_1^+})\\partial_{z_2^+}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-x_2\\rvert^2}\\\\\\nonumber\n&+\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\n&(-\\partial_{x_1^+})\\partial_{x_2^+}(-\\partial_{y_1^+})\\partial_{y_2^+}(-\\partial_{z_1^+})\\partial_{z_2^+}\\frac{1}{\\rvert x_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-x_2\\rvert^2}\\Bigg)\\Bigg\\rvert_{x_1=x_2=x}^{y_1=y_2=y,z_1=z_2=z}\n\\end{align}\nIn the second term of eq. (\\ref{62}) we may conveniently relabel the coordinates, $x_1\\rightarrow x_2$,  $y_1\\rightarrow y_2$,  $z_1\\rightarrow z_2$, and vice versa, because they coincide in the local limit:\n\\begin{align}\n\\nonumber\n\\langle {\\mathbb{O}}_{s_1}(x)&{\\mathbb{O}}_{s_2}(y){\\mathbb{O}}_{s_3}(z)\\rangle =\\frac{1}{(4\\pi^2)^3}\\frac{N^2-1}{8}\\Bigg(\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\\nonumber\n&(-\\partial_{x_1^+})\\partial_{x_2^+}(-\\partial_{y_1^+})\\partial_{y_2^+}(-\\partial_{z_1^+})\\partial_{z_2^+}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-x_2\\rvert^2}\\\\\\nonumber\n&+\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_2^+},\\partial_{x_1^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_2^+},\\partial_{y_1^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_2^+},\\partial_{z_1^+})\\\\\n&(-\\partial_{x_2^+})\\partial_{x_1^+}(-\\partial_{y_2^+})\\partial_{y_1^+}(-\\partial_{z_2^+})\\partial_{z_1^+}\\frac{1}{\\rvert x_2-z_1\\rvert^2}\\frac{1}{\\rvert z_2-y_1\\rvert^2}\\frac{1}{\\rvert y_2-x_1\\rvert^2}\\Bigg)\\Bigg\\rvert_{x_1=x_2=x}^{y_1=y_2=y,z_1=z_2=z}\n\\end{align}\nIn the second term above, we employ the property of the Gegenbauer polynomials (appendix \\ref{appB}):\n\\begin{equation} \\label{GG}\n\\mathcal{G}_{l}^{\\alpha}(\\partial_{x^+_1},\\partial_{x^+_2})=(-1)^l\\mathcal{G}_{l}^{\\alpha}(\\partial_{x^+_2},\\partial_{x^+_1})\n\\end{equation}\nto get:\n\\begin{align}\n\\nonumber\n\\langle {\\mathbb{O}}_{s_1}(x)&{\\mathbb{O}}_{s_2}(y){\\mathbb{O}}_{s_3}(z)\\rangle =\\frac{1}{(4\\pi^2)^3}\\frac{N^2-1}{8}\\Bigg(\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\\nonumber\n&(-\\partial_{x_1^+})\\partial_{x_2^+}(-\\partial_{y_1^+})\\partial_{y_2^+}(-\\partial_{z_1^+})\\partial_{z_2^+}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-x_2\\rvert^2}\\\\\\nonumber\n&+(-1)^{s_1+s_2+s_3}\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\n&(-\\partial_{x_1^+})\\partial_{x_2^+}(-\\partial_{y_1^+})\\partial_{y_2^+}(-\\partial_{z_1^+})\\partial_{z_2^+}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-x_2\\rvert^2}\\Bigg)\\Bigg\\rvert_{x_1=x_2=x}^{y_1=y_2=y,z_1=z_2=z}\n\\end{align}\nTherefore:\n\\begin{align}\n\\label{3point}\n\\nonumber\n\\langle {\\mathbb{O}}_{s_1}(x)&{\\mathbb{O}}_{s_2}(y){\\mathbb{O}}_{s_3}(z)\\rangle =\\frac{1}{(4\\pi^2)^3}(1+(-1)^{s_1+s_2+s_3})\\frac{N^2-1}{8}\\\\\\nonumber\n&\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\n&(-\\partial_{x_1^+})\\partial_{x_2^+}(-\\partial_{y_1^+})\\partial_{y_2^+}(-\\partial_{z_1^+})\\partial_{z_2^+}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-x_2\\rvert^2}\\Bigg\\rvert_{x_1=x_2=x}^{y_1=y_2=y,z_1=z_2=z}\n\\end{align}\nSince the collinear spins are all even, $1+(-1)^{s_1+s_2+s_3}=2$, so that:\n\\begin{align}\n\\label{3point2}\n\\nonumber\n\\langle {\\mathbb{O}}_{s_1}(x)&{\\mathbb{O}}_{s_2}(y){\\mathbb{O}}_{s_3}(z)\\rangle =\\frac{1}{(4\\pi^2)^3}2\\frac{N^2-1}{8}\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\n&(-\\partial_{x_1^+})\\partial_{x_2^+}(-\\partial_{y_1^+})\\partial_{y_2^+}(-\\partial_{z_1^+})\\partial_{z_2^+}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-x_2\\rvert^2}\\Bigg\\rvert_{x_1=x_2=x}^{y_1=y_2=y,z_1=z_2=z}\n\\end{align}\nEq. (\\ref{3point}) also holds for the $3$-point correlators of $\\mathbb{O}_s$, $\\tilde{\\mathbb{O}}_s$, $\\mathbb{S}_s$ and $\\bar{\\mathbb{S}}_s$ below, with the factor of $ 1+(-1)^{s_1+s_2+s_3}$ selecting which of these correlators yield a nonzero result. \nTherefore, by defining:\n\\begin{align}\n\\label{3point1}\n\\nonumber\n& \\mathcal{C}_{s_1s_2s_3}(x,y,z) =\\frac{1}{(4\\pi^2)^3}(1+(-1)^{s_1+s_2+s_3})\\frac{N^2-1}{8}\\\\\\nonumber\n&\\mathcal{G}_{s_1-2}^{\\frac{5}{2}}(\\partial_{x_1^+},\\partial_{x_2^+})\\mathcal{G}_{s_2-2}^{\\frac{5}{2}}(\\partial_{y_1^+},\\partial_{y_2^+})\\mathcal{G}_{s_3-2}^{\\frac{5}{2}}(\\partial_{z_1^+},\\partial_{z_2^+})\\\\\n&(-\\partial_{x_1^+})\\partial_{x_2^+}(-\\partial_{y_1^+})\\partial_{y_2^+}(-\\partial_{z_1^+})\\partial_{z_2^+}\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-x_2\\rvert^2}\\Bigg\\rvert_{x_1=x_2=x}^{y_1=y_2=y,z_1=z_2=z}\n\\end{align}\nthe nonvanishing correlators are:\n\\begin{align}\n\\label{defc3}\n\\langle {\\mathbb{O}}_{s_1}(x){\\mathbb{O}}_{s_2}(y){\\mathbb{O}}_{s_3}(z)\\rangle=\\langle {\\mathbb{O}}_{s_1}(x){\\mathbb{S}}_{s_2}(y)\\bar{\\mathbb{S}}_{s_3}(z)\\rangle = \\mathcal{C}_{s_1s_2s_3}(x,y,z)\n\\end{align}\nand:\n\\begin{align}\n\\label{defc3}\n \\langle {\\mathbb{O}}_{s_1}(x)\\tilde{\\mathbb{O}}_{s_2}(y) \\tilde{\\mathbb{O}}_{s_3}(z)\\rangle = \\mathcal{C}_{s_1s_2s_3}(x,y,z)\n\\end{align}\nAfter substituting the definition of the Gegenbauer polynomials in eq. \\eqref{defg52}, we obtain:\n\\begin{align}\n\\nonumber\n \\mathcal{C}_{s_1s_2s_3}(x,y,z)\n =&\\frac{1}{(4\\pi^2)^3} (1+(-1)^{s_1+s_2+s_3})   \\left(\\frac{2}{4!}\\right)^3\\frac{N^2-1}{8}i^{s_1+s_2+s_3-6}\\\\\\nonumber\n& (s_1+1)(s_1+2)(s_2+1)(s_2+2)(s_3+1)(s_3+2)\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}\\sum_{k_3 = 0}^{s_3-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}{s_3\\choose k_3}{s_3\\choose k_3+2}\\\\\\nonumber\n&(-\\partial_{x_1^+})^{s_1-k_1-1}\\partial^{k_1+1}_{x_2^+}(-\\partial_{y_1^+})^{s_2-k_2-1}\\partial^{k_2+1}_{y_2^+}(-\\partial_{z_1^+})^{s_3-k_3-1}\\partial^{k_3+1}_{z_2^+}\\\\\n&\\frac{1}{\\rvert x_1-y_2\\rvert^2}\\frac{1}{\\rvert y_1-z_2\\rvert^2}\\frac{1}{\\rvert z_1-x_2\\rvert^2}\\Bigg\\rvert_{x_1=x_2=x}^{y_1=y_2=y,z_1=z_2=z}\n\\end{align}\nEmploying eq. \\eqref{doubleder}, we get:\n\\begin{align}\n\\label{3pointO}\n \\mathcal{C}_{s_1s_2s_3}(x,y,z)\n=&-\\frac{1}{(4\\pi^2)^3} (1+(-1)^{s_1+s_2+s_3})   \\left(\\frac{2}{4!}\\right)^3\\frac{N^2-1}{8}i^{s_1+s_2+s_3}2^{s_1+s_2+s_3}\\\\\\nonumber\n& (s_1+1)(s_1+2)(s_2+1)(s_2+2)(s_3+1)(s_3+2)\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1-2}\\sum_{k_2 = 0}^{s_2-2}\\sum_{k_3 = 0}^{s_3-2}{s_1\\choose k_1}{s_1\\choose k_1+2}{s_2\\choose k_2}{s_2\\choose k_2+2}{s_3\\choose k_3}{s_3\\choose k_3+2}\\\\\\nonumber\n&(s_1-k_1+k_2)!(s_2-k_2+k_3)!(s_3-k_3+k_1)!  \\\\\\nonumber\n&\\frac{(x-y)^{s_1-k_1+k_2}_+}{(\\rvert x-y\\rvert^2)^{s_1+1-k_1+k_2}}\\frac{(y-z)^{s_2-k_2+k_3}_+}{(\\rvert y-z\\rvert^2)^{s_2+1-k_2+k_3}}\\frac{(z-x)^{s_3-k_3+k_1}_+}{(\\rvert z-x\\rvert^2)^{s_3+1-k_3+k_1}}\n\\end{align}\nAll the remaining $3$-point correlators vanish. \n\n\\subsection{Extended basis}\n\nThe $3$-point correlators in the extended basis are computed analogously. \\par\nThe nonvanishing correlators are:\n\\begin{align}\n\\label{defc3}\n\\langle {\\mathbb{A}}_{s_1}(x){\\mathbb{A}}_{s_2}(y){\\mathbb{A}}_{s_3}(z)\\rangle=\\langle {\\mathbb{A}}_{s_1}(x){\\mathbb{B}}_{s_2}(y)\\bar{\\mathbb{B}}_{s_3}(z)\\rangle= \\mathcal{A}_{s_1s_2s_3}(x,y,z)\n\\end{align}\nand:\n\\begin{align}\n\\label{defc3}\n\\langle {\\mathbb{A}}_{s_1}(x)\\tilde{\\mathbb{A}}_{s_2}(y)\\tilde{\\mathbb{A}}_{s_3}(z)\\rangle  = \\mathcal{A}_{s_1s_2s_3}(x,y,z)\n\\end{align}\nwith:\n\\begin{align}\n\\nonumber\n \\mathcal{A}_{s_1s_2s_3}(x,y,z)\n =&-\\frac{1}{(4\\pi^2)^3} (1+(-1)^{s_1+s_2+s_3})   \\frac{N^2-1}{8}i^{s_1+s_2+s_3}2^{s_1+s_2+s_3}\\\\\\nonumber\n&\\sum_{k_1 = 0}^{s_1}\\sum_{k_2 = 0}^{s_2}\\sum_{k_3 = 0}^{s_3}{s_1\\choose k_1}{s_1\\choose k_1}{s_2\\choose k_2}{s_2\\choose k_2}{s_3\\choose k_3}{s_3\\choose k_3}\\\\\\nonumber\n&(s_1-k_1+k_2)!(s_2-k_2+k_3)!(s_3-k_3+k_1)!  \\\\\\nonumber\n&\\frac{(x-y)^{s_1-k_1+k_2}_+}{(\\rvert x-y\\rvert^2)^{s_1+1-k_1+k_2}}\\frac{(y-z)^{s_2-k_2+k_3}_+}{(\\rvert y-z\\rvert^2)^{s_2+1-k_2+k_3}}\\frac{(z-x)^{s_3-k_3+k_1}_+}{(\\rvert z-x\\rvert^2)^{s_3+1-k_3+k_1}}\\nonumber \\\\\n\\end{align}\nAll the remaining $3$-point correlators vanish. \n\n\n\\section{$n$-point correlators of twist-$2$ gluonic operators \\label{npoint}}\n\n\nWe compute to the lowest perturbative order $n$-point correlators of the operators in both bases. \\par\n\n\n\n\\subsection{Standard basis}\n\nGiven the bilocal operators:\n\\begin{equation} \\label{tr}\nO(x_i^A,x_i^B) = \\Tr f_{11}(x_i^A) f_{\\dot{1}\\dot{1}}(x_i^B) = \\frac{1}{2} f^a_{11}(x_i^A) f^a_{\\dot{1}\\dot{1}}(x_i^B)\n\\end{equation} \nin the light-cone gauge, the corresponding $n$-point correlator that is connected in the local limit, $x^A_i=x^B_i=x_i$, takes the form:\n\\begin{align}\n\\label{perm0}\n\\nonumber\n\\langle O(x_1^A,x_1^B)\\ldots O(x_n^A,x_n^B) \\rangle=&\\frac{1}{n}\\frac{1}{2^n} \\sum_{i_1 = 1}^n\\ldots \\sum_{i_{n}=1}^n\\Bigg\\rvert_{i_1\\neq i_2\\neq\\ldots\\neq i_{n}}\\hspace{-1.5cm}\\langle f^{a_{i_1}}_{11}(x_{i_1}^A) f^{a_{i_2}}_{\\dot{1}\\dot{1}}(x^B_{i_2})\\rangle\\\\\n&\\langle f^{a_{i_2}}_{11}(x_{i_2}^A) f^{a_{i_3}}_{\\dot{1}\\dot{1}}(x^B_{i_3})\\rangle\\ldots \\langle f^{a_{i_n}}_{11}(x_{i_{n}}^A) f^{a_{i_1}}_{\\dot{1}\\dot{1}}(x^B_{i_1})\\rangle\n\\end{align}\nThe factor of $\\frac{1}{n}$ arises because, if the first index -- for example $i_1=1$ -- is kept fixed, there are only $(n-1)!$ Wick contractions that contribute to the connected correlator.\nA nicer -- but completely equivalent -- formula is written in terms of permutations. If we denote by $P_n$ the set of permutations of $1 \\ldots n$, it follows identically: \n\\begin{align}\n\\nonumber\n\\langle O(x_1^A,x_1^B)\\ldots O(x_n^A,x_n^B)\\rangle =& \\frac{1}{n}\\frac{1}{2^n}\\sum_{\\sigma\\in P_n}\\langle f^{a_{\\sigma(1)}}_{11}(x_{\\sigma(1)}^A) f^{a_{\\sigma(2)}}_{\\dot{1}\\dot{1}}(x^B_{\\sigma(2)})\\rangle\\\\\n&\\langle  f^{a_{\\sigma(2)}}_{11}(x_{\\sigma(2)}^A) f^{a_{\\sigma(3)}}_{\\dot{1}\\dot{1}}(x^B_{\\sigma(3)})\\rangle\\ldots \\langle f^{a_{\\sigma(n)}}_{11}(x_{\\sigma(n)}^A) f^{a_{\\sigma(1)}}_{\\dot{1}\\dot{1}}(x^B_{\\sigma(1)})\\rangle\n\\end{align}\nBesides, eq. \\eqref{axialprop2} reads:\n\\begin{equation}\n\\langle f^a_{11}(x_i) f^b_{\\dot{1}\\dot{1}}(x_j)\\rangle =-\\partial_{x_i^+}\\partial_{x_j^+}\\frac{\\delta^{ab}}{4\\pi^2\\rvert x_i-x_j\\rvert^2}\n\\end{equation}\nHence, for the balanced operators with even $s$, we get in the light-cone gauge:\n\\begin{align}\n\\label{bilinear1}\n\\nonumber\n\\langle \\mathbb{O}_{s_1}(x_1)\\ldots \\mathbb{O}_{s_n}(x_n)\\rangle=&\\frac{1}{2^n}\\mathcal{G}^{\\frac{5}{2}}_{s_1-2}(\\partial_{x_1^{A\\,+}},\\partial_{x_1^{B\\,+}})\\ldots\\mathcal{G}^{\\frac{5}{2}}_{s_n-2}(\\partial_{x_n^{A\\,+}},\\partial_{x_n^{B\\,+}})\\\\\n&\\langle f^{a_1}_{11}(x_1^A)f^{a_1}_{\\dot{1}\\dot{1}}(x_1^B)\\ldots f^{a_n}_{11}(x_n^A){f}^{a_n}_{\\dot{1}\\dot{1}}(x_n^B)\\rangle\\Big\\rvert_{A=B}\n\\end{align}\nwhere:\n\\begin{align} \\label{g}\n\\mathcal{G}^{\\frac{5}{2}}_{s-2}(\\partial_{x_i^+},\\partial_{x_j^+}) &=\\frac{i^{s-2}\\Gamma(3)\\Gamma(s+3)}{\\Gamma(5)\\Gamma(s+1)}\\sum_{k=0}^{s-2} {s\\choose k}{s\\choose k+2}(-1)^{s-k} \\overleftarrow{\\partial}_{x^+_i}^{s-k-2} \\overrightarrow{\\partial}_{x^+_j}^{k}\\nonumber\\\\ &=\\frac{2i^{s-2}(s+1)(s+2)}{4!}\\sum_{k=0}^{s-2} {s\\choose k}{s\\choose k+2}(-1)^{s-k} \\overleftarrow{\\partial}_{x^+_i}^{s-k-2} \\overrightarrow{\\partial}_{x^+_j}^{k}\n\\end{align}\nIt follows from eq. (\\ref{doubleder}) that, correspondingly, the $n$-point correlator contains factors of the form:\n\\begin{align} \\label{no}\n\\nonumber\n&-\\partial_{x_i^+}^{s_i-k_i-2}\\partial_{x_j^+}^{k_j}\\partial_{x_i^+}\\partial_{x_j^+}\\frac{1}{4\\pi^2\\rvert x_i-x_j\\rvert^2} \\\\\n&=\\frac{1}{4\\pi^2}\n (-1)^{s_i-k_i} (s_i-k_i+k_j)!\\,2^{s_i-k_i+k_j} \\frac{(x_i-x_j)_+^{s_i-k_i+k_j}}{(\\rvert x_i-x_j\\rvert^2)^{s_i-k_i+k_j+1}}\n\\end{align}\nTherefore, we get:\n\\begin{align} \\label{000}\n\\nonumber\n&\\langle \\mathbb{O}_{s_1}(x_1)\\ldots \\mathbb{O}_{s_n}(x_n)\\rangle_{conn} =\\frac{(-1)^n}{(4\\pi^2)^n}\\frac{N^2-1}{2^n}i^{\\sum_{l=1}^n s_l}\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma(s_n+3)}{\\Gamma(5)\\Gamma(s_n+1)}\\frac{1}{n}\\sum_{\\sigma\\in P_n}\\\\\\nonumber\n &\\sum_{k_{\\sigma(1)}=0}^{s_{\\sigma(1)}-2}\\ldots\\sum_{k_{\\sigma(n)} = 0}^{s_{\\sigma(n)}-2}{s_{\\sigma(1)}\\choose k_{\\sigma(1)}}{s_{\\sigma(1)}\\choose k_{\\sigma(1)}+2}(-1)^{s_{\\sigma(1)}-k_{\\sigma(1)}}\\ldots {s_{\\sigma(n)}\\choose k_{\\sigma(n)}}{s_{\\sigma(n)}\\choose k_{\\sigma(n)}+2}(-1)^{s_{\\sigma(n)}-k_{\\sigma(n)}}\\\\\\nonumber\n &2^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}(-1)^{s_{\\sigma(1)}-k_{\\sigma(1)}}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\\\\n &\\ldots 2^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}}(-1)^{s_{\\sigma(n)}-k_{\\sigma(n)}}(s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)})!\\frac{(x_{\\sigma(n)}-x_{\\sigma(1)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\nwhere we have set $x^A_i=x^B_i=x_i$ in order to implement the local limit of the bilocal operators. The color factor comes from the contraction of the $n$ Kronecker delta:\n\\begin{equation}\nN^2-1 = \\delta^{a_{\\sigma(1)}a_{\\sigma(2)}}\\delta^{a_{\\sigma(2)}a_{\\sigma(3)}}\\ldots \\delta^{a_{\\sigma(n)}a_{\\sigma(1)}}\n\\end{equation}\nThe overall factor of $(-1)^n$ occurs because of the factor of $i^{-2}$, which is a partial factor of $i^{s-2}$ in eq. \\eqref{g}.\\par\nAfter cancelling between themselves the pairs of factors of the kind $(-1)^{s_a-k_a}$ in eq. \\eqref{000}, and moving out of the sum over the permutations the product of the binomial coefficients, since it is independent of the permutations, we obtain:\n\\begin{align}\n\\label{standardn}\n\\nonumber\n&\\langle \\mathbb{O}_{s_1}(x_1)\\ldots \\mathbb{O}_{s_n}(x_n)\\rangle_{conn} =\\frac{1}{(4\\pi^2)^n}\\frac{N^2-1}{2^n}2^{\\sum_{l=1}^n s_l}i^{\\sum_{l=1}^n s_l}\\\\\\nonumber\n&\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma(s_n+3)}{\\Gamma(5)\\Gamma(s_n+1)}\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_n = 0}^{s_n-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots {s_n\\choose k_n}{s_n\\choose k_n+2}\\\\\\nonumber\n&\\frac{(-1)^n}{n}\\sum_{\\sigma\\in P_n}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n)}-x_{\\sigma(1)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\nActually, if $n$ is even, eq. \\eqref{standardn} also holds for the $n$-point correlator of the operators $\\tilde{\\mathbb{O}}_s$, with the only difference that their collinear spin is odd. \\par\nOtherwise, if $n$ is odd, the correlator vanishes. To verify it, it suffices to notice that, in the sum over the permutations, for every permutation the inverse permutation also occurs with the opposite sign. For example, for $3$-point correlators we get pairs of terms of the kind:\n\\begin{align}\n\\label{sommaperm}\n\\nonumber\n&\\sum_{k_1= 0}^{s_1-2}\\sum_{k_2= 0}^{s_2-2}\\sum_{k_3= 0}^{s_3-2}   \\ldots (s_{1}-k_{1}+k_{2})!(s_{2}-k_{2}+k_{3})!(s_{3}-k_{3}+k_{1})!\\\\\\nonumber\n&(x_1-x_2)^{s_1-k_1+k_2}(x_2-x_3)^{s_2-k_2+k_3}(x_3-x_1)^{s_3-k_3+k_1} \\\\\\nonumber\n&+\\sum_{k_1= 0}^{s_1-2}\\sum_{k_2= 0}^{s_2-2}\\sum_{k_3= 0}^{s_3-2}    \\ldots  (s_{2}-k_{2}+k_{1})!(s_{3}-k_{3}+k_{2})!(s_{1}-k_{1}+k_{3})!\\\\\n&(x_2-x_1)^{s_2-k_2+k_1}(x_3-x_2)^{s_3-k_3+k_2}(x_1-x_3)^{s_1-k_1+k_3} \n\\end{align}\nEmploying the substitution ${k'\\!\\!}_{i} = s_{i}-2-k_{i}$, we obtain for the last term above:\n\\begin{align}\n&\\sum_{k_1= 0}^{s_1-2}\\sum_{k_2= 0}^{s_2-2}\\sum_{k_3= 0}^{s_3-2}   \\ldots (-1)^{s_1+s_2+s_3}(s_{1}-k_{1}+k_{2})!(s_{2}-k_{2}+k_{3})!(s_{3}-k_{3}+k_{1})!\\nonumber \\\\\n&(x_1-x_2)^{s_1-k_1+k_2}(x_2-x_3)^{s_2-k_2+k_3}(x_3-x_1)^{s_3-k_3+k_1}\n\\end{align}\nthat cancels the first term in eq. (\\ref{sommaperm}).\\par\nThe same reasoning applies to the $n+2m+1$-point correlators of balanced operators:\n\\begin{equation}\n\\langle \\mathbb{O}_{s_1}(x_1)\\ldots \\mathbb{O}_{s_{n}}(x_{n})\\tilde{\\mathbb{O}}_{s_{n+1}}(x_{n+1})\\ldots \\tilde{\\mathbb{O}}_{s_{n+2m+1}}(x_{n+2m+1})\\rangle_{conn} = 0\n\\end{equation}\nOtherwise, we get:\n\\begin{align}\n\\label{standardn2}\n\\nonumber\n&\\langle \\mathbb{O}_{s_1}(x_1)\\ldots \\mathbb{O}_{s_n}(x_n)\\tilde{\\mathbb{O}}_{s_{n+1}}(x_{n+1})\\ldots \\tilde{\\mathbb{O}}_{s_{n+2m}}(x_{n+2m})\\rangle_{conn}\\\\\\nonumber\n& =\\frac{1}{(4\\pi^2)^{n+2m}}\\frac{N^2-1}{2^{n+2m}}2^{\\sum_{l=1}^{n+2m} s_l}i^{\\sum_{l=1}^{n+2m} s_l}\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)} \\ldots\\frac{\\Gamma(3)\\Gamma(s_{n+2m}+3)}{\\Gamma(5)\\Gamma(s_{n+2m}+1)}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_{n+2m} = 0}^{s_{n+2m}-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots{s_{n+2m}\\choose k_{n+2m}}{s_{n+2m}\\choose k_{n+2m}+2}\\\\\\nonumber\n&\\frac{(-1)^{n+2m}}{n+2m}\\sum_{\\sigma\\in P_{n+2m}}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n+2m)}-x_{\\sigma(1)})_+^{s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n+2m)}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)}+1}}\n\\end{align}\nFor the correlators of the unbalanced operators, we obtain:\n\\begin{align}\n\\label{bilinear}\n\\nonumber\n&\\langle \\mathbb{S}_{s_1}(x_1)\\ldots \\mathbb{S}_{s_n}(x_n)\\bar{\\mathbb{S}}_{{s'\\!\\!}_1}(y_1)\\ldots \\bar{\\mathbb{S}}_{{s'\\!\\!}_n}(y_n)\\rangle\\\\\\nonumber\n&=\\frac{1}{2^{2n}}\\mathcal{G}^{\\frac{5}{2}}_{s_1-2}(\\partial_{x_1^{A\\,+}},\\partial_{x_1^{B\\,+}})\\ldots\\mathcal{G}^{\\frac{5}{2}}_{s_n-2}(\\partial_{x_n^{A\\,+}},\\partial_{x_n^{B\\,+}})\\mathcal{G}^{\\frac{5}{2}}_{{s'\\!\\!}_1-2}(\\partial_{y_1^{A\\,+}},\\partial_{y_1^{B\\,+}})\\ldots\\mathcal{G}^{\\frac{5}{2}}_{{s'\\!\\!}_n-2}(\\partial_{y_n^{A\\,+}},\\partial_{y_n^{B\\,+}})\\\\\n&\\quad\\frac{1}{2^{n}} \\langle f^{a_1}_{11}(x_1^A)f^{a_1}_{11}(x_1^B)\\ldots f^{a_n}_{11}(x_n^A){f}^{a_n}_{11}(x_n^B) f^{b_1}_{\\dot{1}\\dot{1}}(y_1^A) f^{b_1}_{\\dot{1}\\dot{1}}(y_1^B)\\ldots f^{b_n}_{\\dot{1}\\dot{1}}(y_n^A) f^{b_n}_{\\dot{1}\\dot{1}}(y_n^B)\\rangle\\Big\\rvert_{A=B}\n\\end{align}\nThe factor of $\\frac{1}{2^{2n}}$ arises from the normalization of the color trace in eq. \\eqref{tr}, while the factor of $\\frac{1}{2^n}$ comes from the normalization of the operators.\\par\nWe get the very same correlator by exchanging $A$ and $B$ in all the couples $(x_i^A,x_i^B)$ and $(y_k^A,y_k^B)$ simultaneously:\n\\begin{align}\n\\label{bilinear2}\n\\nonumber\n&\\langle \\mathbb{S}_{s_1}(x_1)\\ldots \\mathbb{S}_{s_n}(x_n)\\bar{\\mathbb{S}}_{{s'\\!\\!}_1}(y_1)\\ldots \\bar{\\mathbb{S}}_{{s'\\!\\!}_n}(y_n)\\rangle\\\\\\nonumber\n&=\\frac{1}{2^n}\\mathcal{G}^{\\frac{5}{2}}_{s_1-2}(\\partial_{x_1^{B\\,+}},\\partial_{x_1^{A\\,+}})\\ldots\\mathcal{G}^{\\frac{5}{2}}_{s_n-2}(\\partial_{x_n^{B\\,+}},\\partial_{x_n^{A\\,+}})\\mathcal{G}^{\\frac{5}{2}}_{{s'\\!\\!}_1-2}(\\partial_{y_1^{B\\,+}},\\partial_{y_1^{A\\,+}})\\ldots\\mathcal{G}^{\\frac{5}{2}}_{{s'\\!\\!}_n-2}(\\partial_{y_n^{B\\,+}},\\partial_{y_n^{A\\,+}})\\\\\n&\\quad\\frac{1}{2^{2n}} \\langle f^{a_1}_{11}(x_1^B)f^{a_1}_{11}(x_1^A)\\ldots f^{a_n}_{11}(x_n^B){f}^{a_n}_{11}(x_n^A) f^{b_1}_{\\dot{1}\\dot{1}}(y_1^B) f^{b_1}_{\\dot{1}\\dot{1}}(y_1^A)\\ldots f^{b_n}_{\\dot{1}\\dot{1}}(y_n^B) f^{b_n}_{\\dot{1}\\dot{1}}(y_n^A)\\rangle\\Big\\rvert_{A=B}\n\\end{align}\nIndeed, in eq. (\\ref{717}) we may conveniently relabel the coordinates, $x_i^A\\rightarrow x_i^B$, $y_k^A\\rightarrow y_k^B$, and vice versa for each $i$ and $k$, since they coincide in the local limit.\nMoreover, according to eq. \\eqref{GG}, $\\mathcal{G}^{\\frac{5}{2}}_{s-2}(\\partial_{x^{A\\,+}},\\partial_{x^{B\\,+}})$ in eqs. \\eqref{bilinear} and \\eqref{bilinear2} is symmetric for the exchange of its arguments, because the collinear spin is even. \\par\nWe evaluate the Wick contractions:\n\\begin{equation}\n\\label{717}\n\\langle f^{a_1}_{11}(x_1^A)\\ldots f^{a_n}_{11}(x_n^A) f^{b_1}_{\\dot{1}\\dot{1}}(y_1^A)\\ldots  f^{b_n}_{\\dot{1}\\dot{1}}(y_n^A)  f^{a_1}_{11}(x_1^B)\\ldots{f}^{a_n}_{11}(x_n^B) f^{b_1}_{\\dot{1}\\dot{1}}(y_1^B)\\ldots f^{b_n}_{\\dot{1}\\dot{1}}(y_n^B)\\rangle\\Big\\rvert_{A=B}\n\\end{equation}\nWe exploit the symmetry above: We only perform the Wick contractions involving the pairing of $x_i^A$ with $y_k^A$ and of $x_i^B$ with $y_k^B$ for any $i,k$, since all the remaining contractions provide the very same result due to the symmetry, and can be taken into account by a symmetry factor that we compute momentarily.\\par\nBesides, since we only are interested in the connected correlator, once $x^A_i$ has been contracted with some $y^A_k$, $x^B_i$ cannot be contracted with $y^B_k$, because the corresponding contribution to the correlator is not connected. \\par\nHence, we construct the correlator as follows: We contract all the $x_i^A$ with the $y_k^A$ and all the $x_{i'}^B$ with the $y_{{k'\\!\\!}}^B$ with $i \\neq i'$ if $k = {k'\\!\\!}$ and $k \\neq {k'\\!\\!}$ if $i = i'$, in such a way that we build a single connected loop. \\par\nThis is realized by summing over two sets of independent permutations arranged in such a way that no disconnected piece may be created: Firstly, we contract $x_{i_1}^A$ with  $y_{k_1}^A$, secondly, we contract $y_{k_1}^B$ with $x_{i_2}^B$ for $i_1\\neq i_2$, then, we contract $x_{i_2}^A$ with $y_{k_2}^A$ for $k_2\\neq k_1$, afterwards, we contract $y_{k_2}^B$ with $x_{i_3}^B$ for $i_3\\neq i_2 \\neq i_1$ and so on, until we arrive at $x_{i_1}^B$, which we contract with the last remaining $y_{k_n}^A$ with $k_n\\neq k_{n-1}\\neq\\ldots\\neq k_1$, in order to close the loop. We end up with a chain that looks like:\n\\begin{align}\n&\\sum_{i_1\\neq i_2\\neq i_3\\ldots\\neq i_n}\\sum_{k_1\\neq k_2\\neq k_3\\ldots\\neq k_n}\\langle f^{a_{i_1}}_{11}(x_{i_1}^A)f^{b_{k_1}}_{\\dot{1}\\dot{1}}(y_{k_1}^A)\\rangle\\langle f^{b_{k_1}}_{\\dot{1}\\dot{1}}(y_{k_1}^B)f^{a_{i_2}}_{11}(x_{i_2}^B)\\rangle\\nonumber\\\\\n&\\langle f^{a_{i_2}}_{11}(x_{i_2}^A)f^{b_{k_2}}_{\\dot{1}\\dot{1}}(y_{k_2}^A)\\rangle\\langle f^{b_{k_2}}_{\\dot{1}\\dot{1}}(y_{k_2}^B)f^{a_{i_3}}_{11}(x_{i_3}^B)\\rangle\\ldots\\langle f^{a_{i_n}}_{11}(x_{i_n}^A) f^{b_{k_n}}_{\\dot{1}\\dot{1}}(y_{k_n}^A)\\rangle\\langle f^{b_{k_n}}_{\\dot{1}\\dot{1}}(y_{k_n}^B)f^{a_{i_1}}_{11}(x_{i_1}^B)\\rangle \n\\end{align}\nHowever, now we are creating a redundancy, since we also are summing on the possible $n$ choices of the starting point of the loop. Therefore, we divide the sum by a factor of $n$:\n \\begin{align}\n &\\frac{1}{n}\\sum_{i_1\\neq i_2\\neq i_3\\ldots\\neq i_n}\\sum_{k_1\\neq k_2\\neq k_3\\ldots\\neq k_n}\\langle f^{a_{i_1}}_{11}(x_{i_1}^A)f^{b_{k_1}}_{\\dot{1}\\dot{1}}(y_{k_1}^A)\\rangle\\langle f^{b_{k_1}}_{\\dot{1}\\dot{1}}(y_{k_1}^B)f^{a_{i_2}}_{11}(x_{i_2}^B)\\rangle\\nonumber\\\\\n &\\langle f^{a_{i_2}}_{11}(x_{i_2}^A)f^{b_{k_2}}_{\\dot{1}\\dot{1}}(y_{k_2}^A)\\rangle\\langle f^{b_{k_2}}_{\\dot{1}\\dot{1}}(y_{k_2}^B)f^{a_{i_3}}_{11}(x_{i_3}^B)\\rangle\\ldots\\langle f^{a_{i_n}}_{11}(x_{i_n}^A) f^{b_{k_n}}_{\\dot{1}\\dot{1}}(y_{k_n}^A)\\rangle\\langle f^{b_{k_n}}_{\\dot{1}\\dot{1}}(y_{k_n}^B)f^{a_{i_1}}_{11}(x_{i_1}^B)\\rangle \n \\end{align}\nA nicer -- but completely equivalent -- formula is written in terms of permutations. \nIt follows identically: \n\\begin{align}\n\\frac{1}{n}\\sum_{\\sigma\\in P_n}\\sum_{\\rho\\in P_n} &\\langle f^{a_{\\sigma_1}}_{11}(x_{\\sigma_1}^A)f^{b_{\\rho_1}}_{\\dot{1}\\dot{1}}(y_{\\rho_1}^A)\\rangle\\langle f^{b_{\\rho_1}}_{\\dot{1}\\dot{1}}(y_{\\rho_1}^B)f^{a_{\\sigma_2}}_{11}(x_{\\sigma_2}^B)\\rangle\\langle f^{a_{\\sigma_2}}_{11}(x_{\\sigma_2}^A)f^{b_{\\rho_2}}_{\\dot{1}\\dot{1}}(y_{\\rho_2}^A)\\rangle\\langle f^{b_{\\rho_2}}_{\\dot{1}\\dot{1}}(y_{\\rho_2}^B)f^{a_{\\sigma_3}}_{11}(x_{\\sigma_3}^B)\\rangle\\nonumber\\\\\n&\\ldots \\langle f^{a_{\\sigma_n}}_{11}(x_{\\sigma_n}^A) f^{b_{\\rho_n}}_{\\dot{1}\\dot{1}}(y_{\\rho_n}^A)\\rangle\\langle f^{b_{\\rho_n}}_{\\dot{1}\\dot{1}}(y_{\\rho_n}^B)f^{a_{\\sigma_1}}_{11}(x_{\\sigma_1}^B)\\rangle \n\\end{align}\nAll the remaining contractions are obtained from this formula by exchanging the coordinates in each couple, $(x_i^A,x_i^B)$ and  $(y_k^A,y_k^B)$, for each $i$ and $k$.\nThere are $2^{2n}$ of such exchanges. \\par\nHowever, the actual degeneration factor is $2^{2n-1}$. Indeed, the extra factor of $\\frac{1}{2}$ comes from the fact that the simultaneous exchange of the coordinates in each couple, $(x_i^A,x_i^B)$ and $(y_k^A,y_k^B)$, yields a contraction that has already been counted due to the symmetry of  eqs. \\eqref{bilinear} and \\eqref{bilinear2} with respect of the simultaneous exchange of $A$ with $B$ in all the coordinate pairs.\\par\nHence, by combining the degeneration factor of $2^{2n-1}$ with the factor of $\\frac{1}{2^n}$ from the normalization of the operators, the overall factor of $2^{n-1}$ survives. It follows:\n\\begin{align}\n\\label{unbalanced}\n\\nonumber\n&\\langle \\mathbb{S}_{s_1}(x_1)\\ldots \\mathbb{S}_{s_n}(x_n)\\bar{\\mathbb{S}}_{{s'\\!\\!}_1}(y_1)\\ldots \\bar{\\mathbb{S}}_{{s'\\!\\!}_n}(y_n)\\rangle=\\frac{1}{(4\\pi^2)^{2n}}\\frac{N^2-1}{2^{2n}}2^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}i^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}\\\\\\nonumber\n&\\frac{\\Gamma(3)\\Gamma(s_1+3)}{\\Gamma(5)\\Gamma(s_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma(s_n+3)}{\\Gamma(5)\\Gamma(s_n+1)}\\frac{\\Gamma(3)\\Gamma({s'\\!\\!}_1+3)}{\\Gamma(5)\\Gamma({s'\\!\\!}_1+1)}\\ldots \\frac{\\Gamma(3)\\Gamma({s'\\!\\!}_n+3)}{\\Gamma(5)\\Gamma({s'\\!\\!}_n+1)}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1-2}\\ldots \\sum_{k_n = 0}^{s_n-2}{s_1\\choose k_1}{s_1\\choose k_1+2}\\ldots {s_n\\choose k_n}{s_n\\choose k_n+2}\\\\\\nonumber\n&\\sum_{{k'\\!\\!}_1=0}^{{s'\\!\\!}_1-2}\\ldots \\sum_{{k'\\!\\!}_n = 0}^{{s'\\!\\!}_n-2}{{s'\\!\\!}_1\\choose {k'\\!\\!}_1}{{s'\\!\\!}_1\\choose {k'\\!\\!}_1+2}\\ldots {{s'\\!\\!}_n\\choose {k'\\!\\!}_n}{{s'\\!\\!}_n\\choose {k'\\!\\!}_n+2}\\\\\\nonumber\n&\\frac{2^{n-1}}{n}\\sum_{\\sigma\\in P_n}\\sum_{\\rho\\in P_n}\n(s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)})!({s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)})!\\\\\\nonumber\n&\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)})!({s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)})!\\\\\\nonumber\n&\\frac{(x_{\\sigma(1)}-y_{\\rho(1)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}}}{\\left(\\rvert x_{\\sigma(1)}-y_{\\rho(1)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}+1}}\\frac{(y_{\\rho(1)}-x_{\\sigma(2)})_+^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}}}{\\left(\\rvert y_{\\rho(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}+1}}\\\\\n&\\ldots\\frac{(x_{\\sigma(n)}-y_{\\rho(n)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}}}{\\left(\\rvert x_{\\sigma(n)}-y_{\\rho(n)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}+1}}\n\\frac{(y_{\\rho(n)}-x_{\\sigma(1)})_+^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}}}{\\left(\\rvert y_{\\rho(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\n\n\n\\subsection{Extended basis}\n\nSimilarly, in the extended basis, we get:\n\\begin{align}\n\\label{bilinear3}\n\\nonumber\n\\langle \\mathbb{A}_{s_1}(x_1)\\ldots \\mathbb{A}_{s_n}(x_n)\\rangle=&\\frac{1}{2^n}\\mathcal{G}^{\\frac{1}{2}}_{s_1}(\\partial_{x_1^{A\\,+}},\\partial_{x_1^{B\\,+}})\\ldots\\mathcal{G}^{\\frac{1}{2}}_{s_n}(\\partial_{x_n^{A\\,+}},\\partial_{x_n^{B\\,+}})\\\\\n&\\langle \\partial_{x_1^{A\\,+}}^{-1}f^{a_1}_{11}(x_1^A)\\partial_{x_1^{B\\,+}}^{-1}f^{a_1}_{\\dot{1}\\dot{1}}(x_1^B)\\ldots \\partial_{x_n^{A\\,+}}^{-1}f^{a_n}_{11}(x_n^A)\\partial_{x_n^{B\\,+}}^{-1}{f}^{a_n}_{\\dot{1}\\dot{1}}(x_n^B)\\rangle\\Big\\rvert_{A=B}\n\\end{align}\nin the light-cone gauge, where:\n\\begin{align} \\label{gext}\n\\mathcal{G}^{\\frac{1}{2}}_{s}(\\partial_{x_1^+},\\partial_{x_2^+}) &=i^{s}\\sum_{k=0}^{s} {s\\choose k}{s\\choose k}(-1)^{s-k} \\overleftarrow{\\partial}_{x^+_1}^{s-k} \\overrightarrow{\\partial}_{x^+_2}^{k}\n\\end{align}\nIt follows from eqs. \\eqref{axialpropeuc2} and \\eqref{doubleder} that, correspondingly, the $n$-point correlator contains factors of the form:\n\\begin{align} \\label{na}\n\\nonumber\n&-\\partial_{x_i^+}^{s_i-k_i}\\partial_{x_j^+}^{k_j}\\frac{1}{4\\pi^2\\rvert x_i-x_j\\rvert^2} \\\\\n&=-\\frac{1}{4\\pi^2}\n(-1)^{s_i-k_i} (s_i-k_i+k_j)!\\,2^{s_i-k_i+k_j} \\frac{(x_i-x_j)_+^{s_i-k_i+k_j}}{(\\rvert x_i-x_j\\rvert^2)^{s_i-k_i+k_j+1}}\n\\end{align}\nTherefore:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{A}_{s_1}(x_1)\\ldots \\mathbb{A}_{s_n}(x_n)\\rangle_{conn} =\\frac{1}{(4\\pi^2)^n}\\frac{N^2-1}{2^n}2^{\\sum_{l=1}^n s_l}i^{\\sum_{l=1}^n s_l}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_n = 0}^{s_n}{s_1\\choose k_1}^2\\ldots {s_n\\choose k_n}^2\\frac{(-1)^n}{n}\\sum_{\\sigma\\in P_n}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n)}-x_{\\sigma(1)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\nwhere now the overall factor of $(-1)^n$ occurs because of the extra minus sign in eq. \\eqref{na} with respect to eq. \\eqref{no}. \\par\nThe very same formula holds for an even number of operators $\\tilde{\\mathbb{A}}_s$, otherwise the correlators vanish. \nWe obtain as well:\n\\begin{align}\n\\nonumber\n&\\langle \\mathbb{A}_{s_1}(x_1)\\ldots \\mathbb{A}_{s_n}(x_n)\\tilde{\\mathbb{A}}_{s_{n+1}}(x_{n+1})\\ldots \\tilde{\\mathbb{A}}_{s_{n+2m}}(x_{n+2m})\\rangle_{conn} \\\\\\nonumber\n&=\\frac{1}{(4\\pi^2)^{n+2m}}\\frac{N^2-1}{2^{n+2m}}2^{\\sum_{l=1}^{n+2m} s_l}i^{\\sum_{l=1}^{n+2m} s_l}\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_{n+2m} = 0}^{s_{n+2m}}{s_1\\choose k_1}^2\\ldots{s_{n+2m}\\choose k_{n+2m}}^2\\\\\\nonumber\n&\\frac{(-1)^{n+2m}}{n+2m}\\sum_{\\sigma\\in P_{n+2m}}(s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)})!\\ldots(s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)})!\\\\\n&\\frac{(x_{\\sigma(1)}-x_{\\sigma(2)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}}}{\\left(\\rvert x_{\\sigma(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+k_{\\sigma(2)}+1}}\\ldots\\frac{(x_{\\sigma(n+2m)}-x_{\\sigma(1)})_+^{s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)}}}{\\left(\\rvert x_{\\sigma(n+2m)}-x_{\\sigma(1)}\\rvert^2\\right)^{s_{\\sigma(n+2m)}-k_{\\sigma(n+2m)}+k_{\\sigma(1)}+1}}\\,\n\\end{align}\nSimilarly, for the unbalanced operators in the extended basis, we get:\n\\begin{align}\n\\label{unbalancedext}\n\\nonumber\n&\\langle \\mathbb{B}_{s_1}(x_1)\\ldots \\mathbb{B}_{s_n}(x_n)\\bar{\\mathbb{B}}_{{s'\\!\\!}_1}(y_1)\\ldots \\bar{\\mathbb{B}}_{{s'\\!\\!}_n}(y_n)\\rangle=\\frac{1}{(4\\pi^2)^{2n}}\\frac{N^2-1}{2^{2n}}2^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}i^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}\\\\\\nonumber\n&\\sum_{k_1=0}^{s_1}\\ldots \\sum_{k_n = 0}^{s_n}\\sum_{{k'\\!\\!}_1=0}^{{s'\\!\\!}_1-2}\\ldots \\sum_{{k'\\!\\!}_n = 0}^{{s'\\!\\!}_n}{s_1\\choose k_1}^2\\ldots {s_n\\choose k_n}^2{{s'\\!\\!}_1\\choose {k'\\!\\!}_1}^2\\ldots {{s'\\!\\!}_n\\choose {k'\\!\\!}_n}^2\\\\\\nonumber\n&\\frac{2^{n-1}}{n}\\sum_{\\sigma\\in P_n}\\sum_{\\rho\\in P_n}\n(s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)})!({s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)})!\\\\\\nonumber\n&\\ldots(s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)})!({s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)})!\\\\\\nonumber\n&\\frac{(x_{\\sigma(1)}-y_{\\rho(1)})_+^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}}}{\\left(\\rvert x_{\\sigma(1)}-y_{\\rho(1)}\\rvert^2\\right)^{s_{\\sigma(1)}-k_{\\sigma(1)}+{k'\\!\\!}_{\\rho(1)}+1}}\\frac{(y_{\\rho(1)}-x_{\\sigma(2)})_+^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}}}{\\left(\\rvert y_{\\rho(1)}-x_{\\sigma(2)}\\rvert^2\\right)^{{s'\\!\\!}_{\\rho(1)}-{k'\\!\\!}_{\\rho(1)}+k_{\\sigma(2)}+1}}\\\\\n&\\ldots\\frac{(x_{\\sigma(n)}-y_{\\rho(n)})_+^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}}}{\\left(\\rvert x_{\\sigma(n)}-y_{\\rho(n)}\\rvert^2\\right)^{s_{\\sigma(n)}-k_{\\sigma(n)}+{k'\\!\\!}_{\\rho(n)}+1}}\n\\frac{(y_{\\rho(n)}-x_{\\sigma(1)})_+^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}}}{\\left(\\rvert y_{\\rho(n)}-x_{\\sigma(1)}\\rvert^2\\right)^{{s'\\!\\!}_{\\rho(n)}-{k'\\!\\!}_{\\rho(n)}+k_{\\sigma(1)}+1}}\n\\end{align}\n\n\n\n\\section{$n$-point correlators and twist-$2$ gluonic operators in Euclidean space-time} \\label{8}\n\n\\subsection{Analytic continuation of $n$-point correlators to Euclidean space-time}\n\n The Minkowskian $n$-point correlators can be analytically continued to Euclidean space-time by substituting (appendix \\ref{appN}):\n \\begin{equation} \\label{ab}\n x_+\\rightarrow -i x_{z}\n \\end{equation}\n and:\n \\begin{equation} \\label{bc}\n \\frac{1}{\\rvert x \\rvert^2}\\rightarrow-\\frac{1}{x^2}\n \\end{equation}\nWe describe the effect of the analytic continuation about various numerical factors.\\par\nIn the standard basis, for the $(n+2m)$-point correlators of balanced operators, the extra factor of $(-i)^{\\sum_{l=1}^{n+2m} s_l}$, which arises from the substitution of eq. \\eqref{ab} into the numerators of eq. \\eqref{O}, cancels out the factor of $i^{\\sum_{l=1}^{n+2m} s_l}$ -- already present in eq. \\eqref{O} -- that comes from the definition of the Minkowskian operators, but the extra factor of $(-1)^{\\sum_{l=1}^{n+2m} (s_l+1)}=(-1)^{\\sum_{l=1}^{n+2m} s_l} (-1)^n$ arises, which comes from the substitution of eq. \\eqref{bc} into the denominators of eq. \\eqref{O}. \nIt combines with the factor of $(-1)^n$ already present in eq. \\eqref{O}, in such a way that only the factor of $(-1)^{\\sum_{l=1}^{n+2m} s_l}$ survives after the analytic continuation.\\par\nSimilarly, for the $2n$-point correlators of unbalanced operators, the factor of  $(-i)^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}$, which arises from the substitution of eq. \\eqref{ab} into the numerators of eq. \\eqref{S}, cancels out the factor of $i^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}$ -- already present in eq. \\eqref{S} -- that comes from the definition of the Minkowskian operators. Thus, only the factor of $(-1)^{\\sum_{l=1}^n (s_l+1+{s'\\!\\!}_l+1)}=(-1)^{\\sum_{l=1}^n s_l+{s'\\!\\!}_l}$, which comes from the substitution of eq. \\eqref{bc} into the denominators of eq. \\eqref{S}, survives after the analytic continuation.\nExactly the same factors survive in the analytic continuation of the corresponding correlators in the extended basis. \\par\n\n\\subsection{Twist-$2$ gluonic operators in Euclidean space-time}\n\n\nAlternatively, the correlators may be computed by first defining the Euclidean operators and afterwards evaluating them in Euclidean space-time.\nOf course, the two procedures must furnish identical results, as we verify momentarily.\\par\nBy performing the Wick rotation (appendix \\ref{appN}) to Euclidean space-time, the operators get rotated as follows.\nThe derivative along the $p_+$ direction transforms as:\n\\begin{equation} \\label{par}\n\\partial_+\\rightarrow i\\partial_{z}\n\\end{equation}\nCorrespondingly, for the elementary operators in the light-cone gauge (appendix \\ref{appC}), we get:\n\\begin{align}\n\\nonumber\n&f_{11}=-\\partial_{+} \\bar{A}\\longrightarrow  f^E_{11}=-i\\partial_{z} \\bar{A}^E\\\\\n&f_{\\dot{1}\\dot{1}}=-\\partial_{+} {A}\\longrightarrow  f_{\\dot{1}\\dot{1}}^E=-i\\partial_{z} {A}^E\n\\end{align}\nand:\n\\begin{align}\n\\nonumber\n&\\partial_+^{-1} f_{11}= -\\bar{A}\\longrightarrow  -i \\partial_z^{-1} f^E_{11}=- \\bar{A}^E\\\\\n&\\partial_+^{-1} f_{\\dot{1}\\dot{1}}=  -{A}\\longrightarrow   -i \\partial_z^{-1} f_{\\dot{1}\\dot{1}}^E= -{A}^E\n\\end{align}\nWe observe that the structure and the sign of the propagators (appendix \\ref{appA1}) of the Euclidean elementary operators, $f^E_{11}, f_{\\dot{1}\\dot{1}}^E$ and $ \\partial_z^{-1} f^E_{11}, \\partial_z^{-1} f_{\\dot{1}\\dot{1}}^E$, are the same as for the Minkowskian operators,  $f_{11}, f_{\\dot{1}\\dot{1}}$ and $ \\partial_+^{-1} f_{11}, \\partial_+^{-1} f_{\\dot{1}\\dot{1}}$, respectively. \\par\nTherefore, the change of the numerical factors in the Euclidean correlators with respect to the Minkowskian correlators may only arise from the change of the numerical factors in the definition of the Euclidean composite operators in terms of the Euclidean elementary operators, $f^E_{11}, f_{\\dot{1}\\dot{1}}^E$ and $ \\partial_z^{-1} f^E_{11}, \\partial_z^{-1} f_{\\dot{1}\\dot{1}}^E$. \\par\nIn the standard basis, we get:\n\\begin{align}\n\\label{basisE}\n\\nonumber\n&\\mathbb{O}_{s}\\rightarrow (-1)^s\\Tr f^E_{11}(x)(\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z})^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_{z}-\\overleftarrow{D}_{z}}{\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z}}\\right) f^E_{\\dot{1}\\dot{1}}(x)=  \\mathbb{O}^E_{s}\\\\\\nonumber\n&\\tilde{\\mathbb{O}}_{s}\\rightarrow (-1)^s\\Tr f^E_{11}(x)(\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z})^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_{z}-\\overleftarrow{D}_{z}}{\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z}}\\right) f^E_{\\dot{1}\\dot{1}}(x)=\\tilde{\\mathbb{O}}^E_{s}\\\\\\nonumber\n&\\mathbb{S}_{s} \\rightarrow\\frac{1}{\\sqrt{2}}(-1)^s\\Tr f^E_{11}(x)(\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z})^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_{z}-\\overleftarrow{D}_{z}}{\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z}}\\right)f^E_{11}(x)=  \\mathbb{S}_{s}^E\\\\\n&\\bar{\\mathbb{S}}_{s} \\rightarrow\\frac{1}{\\sqrt{2}}(-1)^s\\Tr{f}^E_{\\dot{1}\\dot{1}}(x) (\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z})^{s-2}C^{\\frac{5}{2}}_{s-2}\\left(\\frac{\\overrightarrow{D}_{z}-\\overleftarrow{D}_{z}}{\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z}}\\right) f^E_{\\dot{1}\\dot{1}}(x)= \\bar{\\mathbb{S}}^E_{s}\n\\end{align}\nsince the factor of $i^{s-2}$, which comes from the substitution of eq. \\eqref{par} into eq. \\eqref{OO}, combines with the already present factor of $i^{s-2}$ in eq. \\eqref{OO} to produce the factor of $(-1)^{s}$. \\par\nAs a consequence, for all the Euclidean $n$-point correlators in the standard basis, a factor of $i^{\\sum_{l=1}^n (s_l-2)}= i^{\\sum_{l=1}^n s_l}(-1)^n$ disappears with respect to the Minkowskian correlators, because it disappears from the definition of the Euclidean operators, but a factor of $(-1)^{\\sum_{l=1}^n s_l}$ takes its place, in such a way that only the latter survives, thus providing the same result as in the preceding discussion about the analytic continuation. \\par\nSimilarly, in the extended basis, we get:\n\\begin{align}\n\\label{EbasisE}\n\\nonumber\n&\\mathbb{A}_{s}\\rightarrow (-1)^{s+1}\\Tr D_{z}^{-1}f^E_{11}(x)(\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z})^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_{z}-\\overleftarrow{D}_{z}}{\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z}}\\right)D_{z}^{-1} f^E_{\\dot{1}\\dot{1}}(x)= \\mathbb{A}^E_{s}\\\\\\nonumber\n&\\tilde{\\mathbb{A}}_{s}\\rightarrow(-1)^{s+1}\\Tr D_{z}^{-1}f^E_{11}(x)(\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z})^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_{z}-\\overleftarrow{D}_{z}}{\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z}}\\right)D_{z}^{-1} f^E_{\\dot{1}\\dot{1}}(x)= \\tilde{\\mathbb{A}}^E_{s} \\\\\\nonumber\n&\\mathbb{B}_{s} \\rightarrow\\frac{1}{\\sqrt{2}}(-1)^{s+1}\\Tr D_{z}^{-1}f^E_{11}(x)(\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z})^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_{z}-\\overleftarrow{D}_{z}}{\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z}}\\right)D_{z}^{-1}f^E_{11}(x)= \\mathbb{B}_{s}^E\\\\\n&\\bar{\\mathbb{B}}_{s} \\rightarrow\\frac{1}{\\sqrt{2}}(-1)^{s+1}\\Tr D_{z}^{-1} f^E_{\\dot{1}\\dot{1}}(x) (\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z})^{s}C^{\\frac{1}{2}}_{s}\\left(\\frac{\\overrightarrow{D}_{z}-\\overleftarrow{D}_{z}}{\\overrightarrow{D}_{z}+\\overleftarrow{D}_{z}}\\right)D_{z}^{-1} f^E_{\\dot{1}\\dot{1}}(x)=\\bar{\\mathbb{B}}^E_{s}\n\\end{align}\nsince an extra minus sign with respect to the operators in the standard basis comes from the analytic continuation of the two operators $D_+^{-1}$, which contribute $(-i)^2=-1$.\\par\nCorrespondingly, for all the Euclidean $n$-point correlators in the extended basis, a factor of $i^{\\sum_{l=1}^n s_l}$ disappears with respect to the Minkowskian correlators,  because it disappears from the definition of the Euclidean operators, but a factor of $(-1)^{\\sum_{l=1}^n s_l}(-1)^n$ takes its place, which combines with the factor of $(-1)^n$ already present in the Minkowskian correlators, in such a way that only the factor of  $(-1)^{\\sum_{l=1}^n s_l}$ survives, thus providing the same result as in the preceding discussion about the analytic continuation.\\par\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}%\n\\label{sec:introduction}\n\\input{sections\/intro.tex}\n\n\\section{Spatio-temporal Quality Logic}%\n\\label{sec:stql}\n\\input{sections\/stql.tex}\n\n\n\\section{PerceMon: An Online Monitoring Tool}%\n\\label{sec:percemon}\n\\input{sections\/percemon.tex}\n\n\\section{Experiments}%\n\\label{sec:experiments}\n\\input{sections\/experiments.tex}\n\n\\section{Conclusion}%\n\\label{sec:conclusion}\n\nIn this paper, we presented \\emph{PerceMon}, an online monitoring library and tool for\ngenerating monitors for specifications given in Spatio-temporal Quality Logic (STQL).\nWe also present a set of experiments that make use of \\emph{PerceMon}'s integration\nwith the CARLA autonomous car simulator and the ROS middleware platform.\n\nIn future iterations of the tool, we hope to incorporate a more expressive version of\nthe specification grammar that can reason about arbitrary spatial constructs, including\noriented polygons and segmentation regions, and incorporate ways to formally reason\nabout system-level properties (like system warnings and control inputs).\n\n\\section*{Acknowledgment}\n\nThis work was partially supported by the National Science Foundation under grant no.\nNSF-CNS-2038666 and the tool was developed with support from Toyota Research Institute\nNorth America.\n\n\\bibliographystyle{splncs04}\n\n\\subsection{Integration with CARLA and ROS}%\n\\label{sub:integration_with_carla_and_ros}\n\n\nIn this section, we present an integration of the \\emph{PerceMon} tool with the CARLA\nautonomous vehicle simulator~\\cite{dosovitskiy_carla_2017} using the ROS middleware\nplatform~\\cite{quigley_ros_2009}. This follows the example\nof~\\cite{dreossi_verifai_2019} and~\\cite{zapridou_runtime_2020} which interface with\nCARLA, and~\\cite{nickovic_rtamt_2020}, where the tool interfaces with the ROS\nmiddleware platform for use in online monitoring applications.\n\nThe CARLA simulator is an autonomous vehicle simulation environment that uses\nhigh-quality graphics engines to render photo-realistic scenes for testing such\nvehicles. Pairing this with ROS allows us to abstract the data generated by the\nsimulator, the PerceMon monitor, and various perception modules as streams of data or\n\\emph{topics} in a publisher-subscriber network model. Here, a \\emph{publisher}\nbroadcasts data in a known binary format at an endpoint (called a \\emph{topic}) without\nknowing who listens to the data. Meanwhile, a \\emph{subscriber} registers to a specific\ntopic and listens to the data published on that endpoint.\n\nIn our framework, we use the ROS wrapper for\nCARLA\\footnote{\\url{https:\/\/github.com\/carla-simulator\/ros-bridge\/}} to publish all the\ninformation from the simulator, including data from the cameras on the autonomous\nvehicle. The image data is used by perception modules --- like the YOLO object\ndetector~\\cite{redmon_you_2016} and the DeepSORT object\ntracker~\\cite{wojke_simple_2017} --- to publish processed data. The information\npublished by these perception modules can in-turn be used by other perception modules\n(like using detected objects to track them), controllers (that may try to avoid\ncollisions), and by PerceMon online monitors. The architectural overview can be seen\nin~\\autoref{fig:percemon-carla-ros}.\n\nThe use of ROS allows us to reason about data streams independent of the programming\nlanguages that the perception modules are implemented in. For example, the main\nimplementation of the YOLO object detector is written in C\/C++ using a custom deep\nneural network framework called \\emph{Darknet}~\\cite{darknet13}, while the DeepSORT\nobject tracker is implemented in Python. The custom detection formats from each of\nthese algorithms can be converted into standard messages that are published on\npredefined topics, which are then subscribed to from PerceMon. Moreover, this also\npaves the way to migrate and apply PerceMon to any other applications that use ROS for\nperception-based control, for example, in the software stack deployed on real-world\nautonomous vehicles~\\cite{kato_autoware_2018}.\n\n\\section{Semantics for STQL}%\n\\label{sec:semantics_for_stql}\n\nConsider a data stream \\(\\xi\\) consisting of \\emph{frames} containing objects and\nannotated with a time stamp. Let \\(i \\in \\Naturals\\) be the current frame of\nevaluation, and let \\(\\xi_i\\) denote the \\(i^{th}\\) frame. We let \\(\\epsilon: V_t \\cup\nV_f \\to \\Naturals \\cup \\{\\NaN\\} \\) denote a mapping from a pinned time or frame\nvariable to a frame index (if it exists), and let \\(\\zeta: V_o \\to \\Naturals\\) be a\nmapping from an object variable to an actual object ID that was assigned by a\nquantifier. Finally, we let \\(\\Oc(\\xi_i)\\) denote the set of object IDs available in\nthe frame \\(i\\), and let \\(t(\\xi_i)\\) output the timestamp of the given frame.\n\nLet \\(\\Quality{\\varphi}\\) be the quality of the \\stql{} formula, \\(\\varphi\\), at the\ncurrent frame \\(i\\), which can be recursively defined as follows:\n\n\\begin{itemize}\n\n  \\item For the propositional and temporal operations, the semantics simply follows the\n        Boolean semantics for LTL or MTL, i.e.,\n        \\begin{align*}\n          \\Quality{\\top}(\\xi, i, \\epsilon,\n          \\zeta)                                                        & = \\top\n          \\\\\n          \\Quality{\\neg\\varphi}(\\xi, i, \\epsilon, \\zeta)                & = \\neg\\Quality{\\varphi}(\\xi,\n          i, \\epsilon, \\zeta)                                                                                                                          \\\\\n          \\Quality{\\varphi_1 \\lor \\varphi_2}(\\xi, i, \\epsilon,\n          \\zeta)                                                        & = \\Quality{\\varphi_i}(\\xi, i, \\epsilon, \\zeta) \\lor \\Quality{\\varphi_2}(\\xi,\n          i, \\epsilon, \\zeta)                                                                                                                          \\\\\n          \\Quality{\\Next\\varphi}(\\xi, i, \\epsilon, \\zeta)               & =\n          \\Quality{\\varphi}(\\xi, i + 1, \\epsilon, \\zeta)                                                                                               \\\\\n          \\Quality{\\Prev\\varphi}(\\xi, i,\n          \\epsilon, \\zeta)                                              & = \\Quality{\\varphi}(\\xi, i - 1, \\epsilon, \\zeta)                             \\\\\n          \\Quality{\\varphi_1 \\Until \\varphi_2}(\\xi, i, \\epsilon, \\zeta) & = \\bigvee_{i \\leq\n            j}\\left( \\Quality{\\varphi_2}(\\xi, j, \\epsilon, \\zeta) \\land \\bigwedge_{i \\leq k \\leq j}\n          \\Quality{\\varphi_1}(\\xi, k, \\epsilon, \\zeta) \\right)\n          \\\\\n          \\Quality{\\varphi_1 \\Since \\varphi_2}(\\xi, i,\n          \\epsilon, \\zeta)                                              & = \\bigvee_{j \\leq i}\\left(\n          \\Quality{\\varphi_2}(\\xi, j, \\epsilon, \\zeta) \\land \\bigwedge_{j \\leq k \\leq i}\n          \\Quality{\\varphi_1}(\\xi, k, \\epsilon, \\zeta) \\right)\n          \\\\\n        \\end{align*}\n\n  \\item For constraints on time and frame variables,\n        \\begin{align*}\n          \\Quality{x - \\CTIME \\sim c}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\epsilon(x) - t(\\xi_i) \\sim c \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n          \\\\\n          \\Quality{f - \\CFRAME \\sim c}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\epsilon(f) - i \\sim c \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n          \\\\\n        \\end{align*}\n\n  \\item For operations on object variables,\n        \\begin{align*}\n          \\Quality{\\{id_j = id_j\\}}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\zeta(id_j) = \\zeta(id_k) \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n          \\\\\n          \\Quality{\\idcls(id_j) = c}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\Oc(\\xi_i)(\\zeta(id_j)).\\textrm{class} = c \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n          \\\\\n          \\Quality{\\idcls(id_j) = \\idcls(id_k)}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\Oc(\\xi_i)(\\zeta(id_j)).\\textrm{class} \\\\\n            \\qquad\\qquad = \\Oc(\\xi_i)(\\zeta(id_k)).\\textrm{class}       \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n          \\\\\n          \\Quality{\\idprob(id_j) \\sim r}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\Oc(\\xi_i)(\\zeta(id_j)).\\textrm{prob} \\sim r \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n          \\\\\n          \\Quality{\\idprob(id_j) \\sim r \\times \\idprob(id_k)}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\Oc(\\xi_i)(\\zeta(id_j)).\\textrm{prob} \\sim r \\\\\n            \\qquad\\qquad\\times \\Oc(\\xi_i)(\\zeta(id_k)).\\textrm{prob}          \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n        \\end{align*}\n\n  \\item For the area, latitudinal offset, and longitudinal offset,\n        \\begin{align*}\n          \\Quality{\\Area(\\Tc_1) \\sim r}\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\Area(\\SpEval(\\Tc_1, \\xi, \\zeta)) \\sim r \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n          \\\\\n          \\Quality{\\Lat(id_1, \\CRT_1) \\sim r}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } f_{lat}{}(id_1, \\CRT_1, \\xi, i, \\epsilon, \\zeta) \\sim r \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n          \\\\\n          \\Quality{\\Lon(id_1, \\CRT_1) \\sim r}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } f_{lon}(id_1, \\CRT_1, \\xi, i, \\epsilon, \\zeta) \\sim r \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n        \\end{align*}\n        where, \\(\\sim \\in \\Set{<,>,\\leq,\\geq}\\), and\n        \\begin{itemize}\n          \\item \\(f_{lat}\\) computes\n                the \\emph{lateral distance} of the \\(\\CRT\\) point of an object identified by\n                \\(\\Oc(\\zeta(id_1))\\) from the \\emph{Longitudinal axis}; \\item \\(f_{lon}\\) computes the\n                \\emph{longitudinal distance} of the \\(\\CRT\\) point of an object identified by\n                \\(\\Oc(\\zeta(id_1))\\) from the \\emph{Lateral axis}; and \\item \\(\\SpEval(\\Tc, \\xi,\n                \\zeta)\\) is the compound spatial object created after set operations on bounding boxes\n                (defined below).\n        \\end{itemize}\n\n  \\item And, finally, for the spatial existence operator,\n        \\begin{align*}\n          \\Quality{\\SpExists \\Tc}(\\xi, i, \\epsilon, \\zeta)\n           & =\n          \\begin{cases}\n            \\top,\\quad\\text{if } \\SpEval(\\Tc, \\xi, \\zeta) \\not= \\emptyset \\\\\n            \\bot,\\quad\\text{otherwise.}\n          \\end{cases}\n        \\end{align*}\n        Here, the compound spatial function, \\(\\SpEval\\) is defined as follows:\n        \\begin{align*}\n          \\SpEval(\\varnothing, \\xi, \\zeta) & = \\emptyset\n          \\\\\n          \\SpEval(\\Universal, \\xi, \\zeta)  & = \\Universal                                                 \\\\\n          \\SpEval(\\BBox(id), \\xi,\n          \\zeta)                           & = \\zeta(id).\\textrm{bbox}                                    \\\\\n          \\SpEval(\\Comp\\Tc, \\xi, \\zeta)    & =\n          \\Universal \\setminus \\SpEval(\\Tc, \\xi, \\zeta)                                                   \\\\\n          \\SpEval(\\Tc_1 \\sqcup \\Tc_2, \\xi,\n          \\zeta)                           & = \\SpEval(\\Tc_1, \\xi, \\zeta) \\cup \\SpEval(\\Tc_2, \\xi, \\zeta)\n        \\end{align*}\n\n\\end{itemize}\n\n\\subsection{Example 1: Persistent Detection}%\n\\label{sub:example1}\n\nIn this example, we are interested in monitoring the following high-level specification:\n\\begin{displayquote}\n  If an object is detected in the current image with high confidence and away from the\n  margins, then it must have been present in the previous set of frames too.\n\\end{displayquote}\nIn this specification, we are essentially interested in tracking is there is a span of\ntime when we lose track of an object that is in the current frame. Formally, the\nspecification is as follows:\n\\begin{align}\n  \\begin{split}\n    \\phi &:= \\forall\\Set{id_1} @ \\Set{f}.\\left( \\left(\\phi_{\\text{high prob}} \\land \\phi_{\\text{margins}}\\right) \\Rightarrow \\Alw\\phi_{\\text{exists}}\\right) \\\\[1em]\n    \\phi_{\\text{high prob}} &:= \\idprob(id_1) > 0.8 \\land\\idcls(id_1) = \\textsf{PED} \\\\\n    \\phi_{\\text{margins}} &:= \\Lon(id_1, \\TM) > c_1 \\land \\Lon(id_1, \\BM) < c_2 \\\\\n    &\\quad\\land \\Lat(id_1, \\LM) > c_3 \\land \\Lat(id_1, \\RM) < c_4 \\\\\n    \\phi_{\\text{exists}} &:= (f - \\CFRAME < 6) \\\\\n    &\\quad \\Rightarrow \\exists\\{id_2\\}.\\left(\\{id_1 = id_2\\} \\land \\idprob(id_2) > 0.7 \\land \\idcls(id_2) = \\textsf{PED}\\right)\n  \\end{split}\n\\end{align}\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example1\/00022.jpg}\n    \\caption{Frame 22.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example1\/00023.jpg}\n    \\caption{Frame 23.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example1\/00024.jpg}\n    \\caption{Frame 24.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example1\/00025.jpg}\n    \\caption{Frame 25.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example1\/00026.jpg}\n    \\caption{Frame 26.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example1\/00027.jpg}\n    \\caption{Frame 27.}\n  \\end{subfigure}\n\n  \\caption{A snippet from the video \\texttt{MOT17--03}. Here, it can be seen that the\n  bounding boxes around the pedestrians in the frame are being lost. The perception\n  monitor is able to detect these lost frames.}%\n\\end{figure}\n\n\n\\subsection{Example 2: Potential Occlusions}%\n\\label{sub:example1}\n\nIn this example, we are interested in monitoring the following high-level specification:\n\\begin{displayquote}\n  If an object is detected in the current image with high confidence and away from the\n  margins, it should not have been overlapping with another object too much.\n\\end{displayquote}\nIn this specification, we are essentially interested in checking if any object was too\nclose to another object, and could have potentially occluded the other object. Formally,\nthe specification is as follows:\n\\begin{align}\n  \\begin{split}\n   \\phi &:= \\forall\\Set{id_1} @ \\Set{x}.\\left( \\left(\\phi_{\\text{high prob}} \\land \\phi_{\\text{margins}}\\right)\\right. \\\\\n   &\\qquad\\qquad\\qquad\\left. \\Rightarrow \\Alw(x - \\CTIME < 1.0 \\Rightarrow \\neg\\phi_{\\text{occluding}}\\right) \\\\[1em]\n   \\phi_{\\text{high prob}} &:= \\idprob(id_1) > 0.8 \\\\\n   \\phi_{\\text{margins}} &:= \\Lon(id_1, \\TM) > c_1 \\land \\Lon(id_1, \\BM) < c_2 \\\\\n                         &\\quad\\land \\Lat(id_1, \\LM) > c_3 \\land \\Lat(id_1, \\RM) < c_4 \\\\\n   \\phi_{\\text{occluding}} &:=\\exists\\{id_2\\}.\\left(\\{id_1 = id_2\\} \\land \\idprob(id_2) > 0.7\\right.\\\\\n   &\\quad\\left.\\land \\forall\\{id_3\\} . \\right(\\{id_3 \\not= id_2\\} \\Rightarrow (\\Area(\\BBox(id_3) \\sqcap \\BBox(id_2)) > a_1) \\left)\\right)\n  \\end{split}\n\\end{align}\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example2\/00122.jpg}\n    \\caption{Frame 122.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example2\/00123.jpg}\n    \\caption{Frame 123.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example2\/00124.jpg}\n    \\caption{Frame 124.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example2\/00125.jpg}\n    \\caption{Frame 125.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example2\/00126.jpg}\n    \\caption{Frame 126.}\n  \\end{subfigure}\n  ~%\n  \\begin{subfigure}[t]{0.15\\textwidth}\n    \\centering\n    \\includegraphics[height=2cm,keepaspectratio]{imgs\/Example2\/00127.jpg}\n    \\caption{Frame 127.}\n  \\end{subfigure}\n\n  \\caption{A snippet from the video \\texttt{MOT17--03}. Here, it can be seen that the\n  bounding boxes around the pedestrians in the frame are being lost due to various\n  reasons, including occlusions. Our perception monitor is able to detect these\n  occlusions before it happens.}%\n\\end{figure}\n\n\n\n\n\\subsection{Spatio-temporal Quality Logic}%\n\nSpatio-temporal quality logic (STQL)~\\cite{hekmatnejad_formalizing_2021} is an\nextension of Timed Quality Temporal Logic (TQTL)~\\cite{dokhanchi_evaluating_2018} that\nincorporates reasoning about high-level topological structures present in perception\ndata, like bounding boxes, and set operations over these structures.\n\nSTL has been used extensively in testing and monitoring of control systems mainly due\nto the ability to express rich specifications on low-level, real-valued signals\ngenerated from these systems. To make the logic more high-level, spatial extensions\nhave been proposed that are able to reason about spatial relations between\nsignals~\\cite{bortolussi_specifying_2014,nenzi_qualitative_2015,gabelaia_computational_2003,haghighi_spatel_2015}.\nA key feature of data streams generated by perception algorithms is that they contain\n\\emph{frames} of  spatial objects consisting of both, real-values and discrete-valued\nquantities: the discrete-valued signals are the IDs of the objects and their associated\ncategories; while real-valued signals include bounding boxes describing the objects and\nconfidence associated with their identities. While STL and MTL can be used to reason\nabout properties of a fixed number of such objects in each frame by creating signal\nvariables to encode each of these properties, it is not possible to design monitors\nthat handle arbitrarily many objects per frame.\n\nTQTL~\\cite{dokhanchi_evaluating_2018} is a logic that is specifically catered for\nspatial data from perception algorithms. Using Timed Propositional Temporal\nLogic~\\cite{bouyer_expressiveness_2005} as a basis, TQTL allows one to \\emph{pin} or\n\\emph{freeze} the signal at a certain time point and use clock variables associated\nwith the freeze operator to define time constraints. Moreover, TQTL introduces a\nquantifier over objects in a frame and the ability to refer to properties intrinsic to\nthe object: tracking IDs, classes or categories, and detection confidence.\nSTQL~\\cite{hekmatnejad_formalizing_2021} further extends the logic to reason about the\nbounding boxes associated with these objects, along with predicate functions for these\nspatial sets, by incorporating topological semantics from the \\(S4_u\\) spatio-temporal\nlogic~\\cite{gabelaia_computational_2003}.\n\n\\begin{definition}[STQL\n    Syntax~\\cite{hekmatnejad_formalizing_2021}]%\n  \\label{def:stql-syntax} Let \\(V_t\\) be a set of time variables, \\(V_f\\) be a set of\n  frame variables, and \\(V_o\\) be a set of object ID variables. Then the syntax for STQL\n  is recursively defined by the following grammar:\n  \\begin{align*}\n   \n    \\varphi ::=\\quad%\n                    & \\exists\\{ {id}_1, {id}_2, \\ldots\\}@\\varphi \\mid\n    \\{x,f\\}.\\varphi                                                         \\\\\n                    & \\mid \\top \\mid \\neg\\varphi \\mid \\varphi \\lor \\varphi\n    \\mid \\Next\\varphi \\mid \\Prev\\varphi \\mid \\varphi \\Unt \\varphi \\mid \\varphi\n    \\Since \\varphi                                                          \\\\\n                    & \\mid \\CTIME - x \\sim t \\mid \\CFRAME - f \\sim n\n    \\\\\n                    & \\mid \\idcls({id}_i) = c \\mid \\idcls({id}_i) =\n    \\idcls({id}_i)\n    \\mid \\idprob({id}_i) \\geq r \\mid \\idprob({id}_i) \\geq r \\times\n    \\idprob({id}_j)                                                         \\\\\n                    & \\mid \\{{id}_i = {id}_j\\} \\mid \\{{id}_i \\not= {id}_j\\}\n    \\mid \\SpExists \\Omega \\mid \\Pi                                          \\\\\n    \\Omega ::=\\quad & \\varnothing \\mid \\Universal \\mid \\BBox({id}_1)\n    \\mid \\Comp\\Omega \\mid \\Omega \\sqcup \\Omega\n    \\\\\n    \\Pi ::=\\quad    & \\Area(\\Omega) \\geq r | \\Area(\\Omega) \\geq r \\times\n    \\Area(\\Omega)                                                           \\\\\n                    & \\mid \\ED({id}_i, \\CRT, {id}_j, \\CRT) \\geq r\n    \\mid \\Theta \\geq r \\mid \\Theta \\geq r \\times \\Theta\n    \\\\\n   \n    \\Theta ::=\\quad & \\Lat({id}_i, \\CRT) \\mid \\Lon({id}_i, \\CRT)\n    \\\\\n   \n    \\CRT ::=\\quad   & \\LM \\mid \\RM \\mid \\TM \\mid \\BM \\mid \\CT\n  \\end{align*}\n\n  Here, \\({id}_i \\in V_o\\) (for all indices \\(i\\)), \\(x \\in V_t\\), and \\(f \\in V_f\\). In\n  the above grammar \\(r\\) is a real-valued constant that allows for the comparison of\n  ratios of object properties.\n\\end{definition}\n\nIn the above grammar, \\(\\neg\\varphi\\) and \\(\\varphi \\lor \\varphi\\) are, respectively,\nthe negation and disjunction operators from propositional logic while \\(\\Next\\varphi\\),\n\\(\\Prev\\varphi \\), \\( \\varphi \\Unt \\varphi \\), and \\( \\varphi \\Since \\varphi\\) are the\ntemporal operators \\emph{next}, \\emph{previous}, \\emph{until}, and \\emph{since}\nrespectively. The above grammar can be further used to derive the other propositional\noperators, like conjunction (\\(\\varphi \\land \\varphi\\)), along with temporal operators\nlike \\emph{always} (\\(\\Alw \\varphi\\)) and \\emph{eventually} (\\(\\Ev \\varphi\\)), and\ntheir past-time equivalents \\emph{holds} (\\(\\AlwP \\varphi\\)) and \\emph{once} (\\(\\EvP\n\\varphi\\)). In addition to that, STQL extends these by introducing freeze quantifiers\nover clock variables and object variables. \\(\\Set{x, f}.\\varphi\\) freezes the time and\nframe that the formula \\(\\varphi\\) is evaluated, and assigns them to the clock\nvariables \\(x\\) and \\(f\\), where \\(x\\) refers to pinned time variables and \\(f\\) refers\nto pinned frame variables. The constants, \\(\\CTIME, \\CFRAME\\) refer to the value of the\ntime and frame number where the current formula is being evaluated. This allows for the\nexpression \\(x - \\CTIME\\) and \\(f - \\CFRAME\\) to measure the duration and the number of\nframes elapsed, respectively, since the clock variables \\(x\\) and \\(f\\) were pinned.\nThe expression \\(\\exists\\Set{{id}_1}@\\varphi\\) searches over each object in a frame in\nthe incoming data stream --- assigning each object to the object variable \\({id}_1\\)\n--- if there exists an object that satisfies \\(\\varphi\\). The functions \\(\\idcls(id)\\)\nand \\(\\idprob(id)\\) refer to the \\emph{class} and \\emph{confidence} the detected object\nassociated with the ID variable. In addition to these TQTL operations, bounding boxes\naround objects can be extracted using the expression \\(\\BBox(id)\\) and set topological\noperations can be defined over them. The \\emph{spatial exists} operator\n\\(\\SpExists\\Omega\\) checks if the spatial expression \\(\\Omega\\) results in a non-empty\nspace or not. Quantitative operations like \\(\\Area(\\cdot)\\) measure the area of spatial\nsets; \\(\\ED\\) computes the Euclidean distances between references points (\\(\\CRT\\)) of\nbounding boxes; and \\(\\Lat\\) and \\(\\Lon\\) measure the latitudinal and longitudinal\noffset of bounding boxes respectively. Here, \\(\\CRT\\) refers to the reference points\n--- left, right, top, and bottom margins, and the centroid --- for bounding boxes. Due\nto lack of space, we defer defining the formal semantics of STQL\nto~\\refappendix{sec:semantics_for_stql} and also refer the readers\nto~\\cite{hekmatnejad_formalizing_2021} for more extensive details.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCorrupted speech signal by noise and reverberation is one of the most common signals we hear in our everyday life.\nThe desire to listen to clean speech signal, therefore, is strong let alone its usage for machine such as speech recognition system.\nWhile there has been numerous studies to address the problem of single channel denoising and dereverberation, only few have tried to solve this problem with a single deep learning model \\cite{sun2018enhanced}.\nThis motivates us to tackle this real-world problem using a single-stage deep learning model.\nWe address this problem by dissecting the elements that compose the noisy-reverberant mixture, that is, noise, direct source, and reverberation.\nBy dissecting each part of the mixture, one could handle each element at hand and even mix them with a desired proportion.\nNote that this is a desirable property for users as the effective amount of reverberation is important to achieve better speech intelligibility for both impaired and nonimpaired listeners \\cite{bradley2003importance, hu2014effects}.\n\nThe key contributions of our proposed approach are three folds.\nFirst, to suppress the noise and reverberation, we propose a new type of complex-valued mask called phase-aware $\\beta$-sigmoid mask (PHM).\nWhile the complex-valued mask suggested by \\cite{williamson2016complex} separately estimates the real and imaginary part of a complex spectrogram, we believe the phase part of it can be effectively estimated by reusing an estimated magnitude value of it in a trigonometric perspective as suggested in \\cite{wang2019masking}.\nThe major difference between PHM and the suggested approach in \\cite{wang2019masking} is that PHM is designed to respect the triangular relationship between mixture, source and the rest, and hence the sum of the estimated source and the rest is always equal to the mixture.\nBy exploiting this property, we train the deep network to output two different PHMs simultaneously to effectively deal with both denoising and derverbration problem.\nSecond, we propose a new time-domain loss function, an emphasized multi-scale cos similarity loss function.\nA time-domain loss function has recently been used as a popular loss function \\cite{choi2019phase, yao2019coarse, wang2018end, koizumi2020speech, le2019sdr}.\nTo better design the time-domain cos similarity loss function proposed in \\cite{choi2019phase}, we change it into a multi-scale version of it with proper emphasis functions and show the effectiveness of it.\nFinally, we suggest an optimization strategy for two-dimensional U-Net to significantly reduce the computational inefficiency in runtime.\n\n\\section{Related Works}\nRecently, there has been an increasing interests in phase-aware speech enhancement because of the sub-optimality of reusing the phase of mixture signal.\nThe first work that tried to address this problem was by using phase-sensitive mask (PSM) \\cite{erdogan2015phase}. PSM estimates the real-part of the signal which is still sub-optimal.\nAs a more direct remedy for this, a complex masking \\cite{williamson2016complex, choi2019phase, wang2019masking} or complex spectral mapping \\cite{tan2019complex} has also been proposed to estimate a clean phase part.\nAnother line of research is to sequentially estimate the clean phase part using an additional sub-module \\cite{takahashi2018phasenet, afouras2018conversation, yin2019phasen}. This, however, is limited in that it requires an additional module resulting in inefficient computation.\nWhile most of these works tried to estimate the clean phase by using phase mask or an additional network, the absolute phase difference between mixture and source can be actually computed using the law of cosines using the estimated magnitude values as the three sides of a triangle \\cite{mowlaee2012phase, mowlaee2014time}.\nInspired by this, \\cite{wang2019deep} proposed to estimate a rotational direction of the absolute phase difference using a sign-prediction network.\n\nThe efforts to deal with denoising or dereverberation using deep networks have been tried in many works. Recently, \\cite{zhao2018two, maciejewski2019whamr} tried to address this problem with two modules for each task. \nWe believe, however, a two-stage framework is not necessary and can be achieved using a single deep network.\n\n\\section{Single-stage Denoising and Dereverberation}\nA noisy-reverberant mixture signal $\\bm{x}$ is commonly modeled as the sum of additive noise $\\bm{y}^{(n)}$ and reverberant source $\\tilde{\\bm{y}}$, where $\\tilde{\\bm{y}}$ is a result of convolution between room impulse response (RIR) $\\bm{h}$ and dry source $\\bm{y}$ as follows,\n$\\bm{x} = \\tilde{\\bm{y}} + \\bm{y}^{(n)} = \\bm{h} \\circledast \\bm{y} + \\bm{y}^{(n)}$.\nMore concretely, we can break down $\\bm{h}$ into two parts, that is, direct path part $\\bm{h}^{(d)}$ which does not includes the reflection path and the rest of the part $\\bm{h}^{(r)}$ that includes all the reflection paths as follows, \n$\\bm{x} = (\\bm{h}^{(d)} + \\bm{h}^{(r)}) \\circledast \\bm{y} + \\bm{y}^{(n)} = \\bm{h}^{(d)} \\circledast \\bm{y} + \\bm{h}^{(r)} \\circledast \\bm{y} + \\bm{y}^{(n)} = \\bm{y}^{(d)} + \\bm{y}^{(r)} + \\bm{y}^{(n)}$,\nwhere $\\bm{y}^{(d)}$ and $\\bm{y}^{(r)}$ denotes direct path source and reverberation, respectively.\nIn this setting, our goal is to separate $\\bm{x}$ into three elements $\\bm{y}^{(d)}$, $\\bm{y}^{(r)}$, and $\\bm{y}^{(n)}$.\nEach of the corresponding time-frequency $(t,f)$ representations computed by STFT is denoted as $X_{t,f} \\in \\mathbb{C}$, $Y^{(d)}_{t,f}\\in \\mathbb{C}$, $Y^{(r)}_{t,f} \\in \\mathbb{C}$, $Y^{(n)}_{t,f} \\in \\mathbb{C}$, and the estimated values will be denoted by the hat operator $\\hat{\\, \\cdot \\,}$. \n\n\\subsection{Phase-aware \\texorpdfstring{$\\beta$}{}-sigmoid mask}\nDesigning a mask that is not limited to output the optimal value of ideal mask requires two conditions to satisfy.\nFirst, the range of magnitude mask should not be limited.\nSecond, the mask has to be complex-valued so that it can correct both the magnitude part and phase part of the mixture signal.\nThe proposed phase-aware $\\beta$-sigmoid mask (PHM) is designed to handle both conditions while systemically restricting the sum of estimated complex values to be exactly the value of mixture, $X_{t,f} = Y^{(k)}_{t,f} + Y^{(\\lnot k)}_{t,f}$.\nThe PHM separates the mixture $X_{t,f}$ in STFT domain into two parts as \\textit{one-vs-rest} approach, that is, the signal $Y^{(k)}_{t,f}$ and the sum of the rest of the signals $Y^{(\\lnot k)}_{t,f} = X_{t,f}-Y^{(k)}_{t,f}$, where index $k$ could be one of direct path source ($d$), reverberation ($r$), and noise ($n$) in our setting, $k \\in \\{d, r, n\\}$.\nThe complex-valued mask $M^{(k)}_{t,f} \\in \\mathbb{C}$ estimates the magnitude and phase value of the source of interest $k$.\nThe mask is composed of two parts, (1) magnitude mask estimation, (2) phase estimation by reusing the magnitude estimation from (1) and two-class sign prediction.\n\nFirst, the network outputs the magnitude part of two masks $\\abs{M^{(k)}_{t,f}}$ and $\\abs{M^{(\\lnot k)}_{t,f}}$ with sigmoid function $\\sigma^{(k)}(\\bm{z}_{t,f})$ multiplied by coefficient $\\beta_{t,f}$ as follows,\n\\begin{equation}\n\\label{eq:magnitude_mask}\n\\abs{M^{(k)}_{t,f}} = \\beta_{t,f} \\cdot \\sigma^{(k)}(\\bm{z}_{t,f}) = \\beta_{t,f} \\cdot \\frac{1}{1+e^{-(z^{(k)}_{t,f} - z^{(\\lnot k)}_{t,f})}}\n\\end{equation}\nwhere $z^{(k)}_{t,f}$ is the output located at $(t,f)$ from the last layer of neural-network function $\\psi^{(k)}(\\phi)$, and $\\phi$ is a function composed of network layers before the last layer.\n$\\abs{M^{(k)}_{t,f}}$ serves as a magnitude mask to estimate source $k$ and the value of it ranges from 0 to $\\beta_{t,f}$.\nThe role of $\\beta_{t,f}$ is to design a mask that is close to the optimal mask with a flexible magnitude range so that the value is not bounded between 0 and 1 unlike the typically used sigmoid mask.\nIn addition, because the sum of the complex valued masks $M^{(k)}_{t,f}$ and $M^{(\\lnot k)}_{t,f}$ must compose a triangle, it is reasonable to design a mask that satisfies the triangle inequalities, that is, $\\abs{M^{(k)}_{t,f}} + \\abs{M^{(\\lnot k)}_{t,f}}$ $\\geq 1$  and $\\abs{\\, \\abs{M^{(k)}_{t,f}} - \\abs{M^{(\\lnot k)}_{t,f}}} \\leq 1$.\nTo address the first inequality we designed the network to output $\\beta_{t,f}$ from the last layer with a softplus activation function as follows, $\\beta_{t,f} = 1+ \\texttt{softplus}((\\psi_{\\beta}(\\phi))_{t,f})$, where $\\psi_{\\beta}$ denotes an additional network layer to output $\\beta_{t,f}$. The second inequality can be satisfied by clipping the upper bound of the $\\beta_{t,f}$ by $1 \\mathbin{\/} \\abs{\\, \\sigma^{(k)}(\\bm{z}_{t,f}) - \\sigma^{(\\lnot k)}(\\bm{z}_{t,f})}$.\n\nOnce the magnitude masks are decided, we can construct a phase mask $e^{j\\theta_{t,f}^{(k)}}$. \nGiven the magnitudes as three sides of a triangle, we can compute the cosine of absolute phase difference $\\Delta \\theta_{t,f}^{(k)}$ between the mixture and source $k$ as follows,\n$\\cos(\\Delta \\theta_{t,f}^{(k)}) = \\nicefrac{ (1+\\abs{M_{t,f}^{(k)}}^2 - \\abs{M_{t,f}^{(\\lnot k)}}^2 ) } {(2 \\, \\abs{M_{t,f}^{(k)}})}$.\nNext, the rotational direction (clockwise or counterclockwise) for phase correction can be decided by estimating sign value $\\xi_{t,f} \\in \\{-1, 1\\}$ as follows,\n\\begin{equation}\n\\label{eq:phase_mask}\ne^{j\\theta_{t,f}^{(k)}} = \\cos(\\Delta \\theta_{t,f}^{(k)}) + j \\xi_{t,f}\\sin(\\Delta \\theta_{t,f}^{(k)}).\n\\end{equation}\nTwo-class straight-through Gumbel-softmax estimator was used to estimate $\\xi_{t,f}$ \\cite{jang2016categorical}. It allows us to discretize the output of the Gumbel-softmax function $\\gamma^{(i)}$ with $\\argmax$ and still be able to train the network in an end-to-end manner using a continuous approximation in the backward pass.\n$\\xi_{t,f}$ is defined as follows,\n\\begin{equation}\n\\label{eq:sign}\n\\xi_{t,f} = \n    \\begin{cases}\n        -1, & \\gamma^{(0)}(\\bm{q}_{t,f}) > \\gamma^{(1)}(\\bm{q}_{t,f})\\\\\n        1, & \\text{otherwise}\n    \\end{cases}\n\\end{equation}\nwhere $\\gamma^{(i)}(\\bm{q}_{t,f})$ is defined as follows,\n\\begin{equation}\n\\label{eq:gumbel}\n\\gamma^{(i)}(\\bm{q}_{t,f})=\\frac{e^{q^{(i)}_{t,f}}}{\\sum_{i}{e^{q^{(i)}_{t,f}}}} = \\frac{e^{((\\psi_{i}(\\phi))_{t,f} + g_i)\\mathbin{\/}\\tau}}{\\sum_{i}{e^{((\\psi_{i}(\\phi))_{t,f} + g_i)\\mathbin{\/}\\tau}}},\n\\end{equation}\nand $g_0$ and $g_1$ are samples from Gumbel$\\left(0, 1\\right)$, $\\psi_i$ is an additional network layer to output logit value $q^{(i)}_{t,f}$, and $\\tau$ is a temperature parameter for Gumbel-softmax.\nFinally, $M^{(k)}_{t,f}$ is defined as follows,\n\\begin{equation}\nM^{(k)}_{t,f} = \\abs{M^{(k)}_{t,f}}e^{j\\theta_{t,f}^{(k)}}.\n\\end{equation}\n\n\\subsection{Masking from the perspective of quadrangle}\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.4]{figs\/quadrangle.png}\n\\caption{The illustration of masks on quadrangle}\n\\label{fig:quadrangle}\n\\end{figure}\nAs we desire to extract both direct source and reverberant source, two pairs of PHMs are used for each of them.\nThe first pair of masks separates direct source and the rest of the component, denoted as $M^{(d)}_{t,f}$ and $M^{(\\lnot d)}_{t,f}$.\nThe second pair of masks separates noise and reverberant source component, denoted as $M^{(n)}_{t,f}$ and $M^{(\\lnot n)}_{t,f}$.\nSince PHM guarantees the mixture and separated components to construct a triangle in the complex STFT domain, the outcome of the separation can be seen from the perspective of quadrangle as in Fig \\ref{fig:quadrangle}.\nIn this setting, as the three sides and two side angles are already determined by the two pairs of PHMs, the last fourth side of quadrangle, the reverberation component $\\hat{Y}^{(r)}_{t,f}$, is uniquely decided.\n\n\\subsection{Emphasized multi-scale cosine similarity loss}\nLearning to maximize cosine similarity can be regarded as maximizing the signal-to-distortion ratio (SDR) \\cite{choi2019phase}. Cosine similarity loss $C$ between estimated signal $\\hat{\\bm{y}}^{(k)} \\in \\mathbb{R}^{N} $ and ground truth signal $\\bm{y}^{(k)} \\in \\mathbb{R}^{N}$  is defined as follows,\n\\begin{equation}\n\\label{eq:cossim}\n\tC(\\bm{y}^{(k)},\\hat{\\bm{y}}^{(k)}) = -\\frac{<\\bm{y}^{(k)},\\hat{\\bm{y}}^{(k)}>}{\\norm{\\bm{y}^{(k)}}  \\norm{\\hat{\\bm{y}}^{(k)}}},\n\\end{equation}\nwhere $N$ denotes the temporal dimensionality of a signal and $k$ denotes the type of signal ($k \\in \\{d,r,n\\}$).\nConsider a sliced signal $\\bm{y}^{(k)}_{[\\frac{N}{M}(i-1):\\frac{N}{M}i]}$, where $i$ denotes the segment index and $M$ denotes the number of segment.\nBy slicing the signal and normalize it by its norm, each sliced segment is considered as an unit for computing $C$.\n Therefore, we hypothesize that it is important to choose a proper segment length unit $\\frac{N}{M}$ when computing $C$.\n In our case, we used multiple settings of segment lengths $g_{j}=\\frac{N}{M_{j}}$ as follows,\n\\begin{equation}\n\\label{eq:multiscale}\n\t\\mathcal{L}(\\bm{y}^{(k)},\\hat{\\bm{y}}^{(k)}) = \\sum_{j} \\frac{1}{M_j}\\sum_{i=1}^{M_j} C(\\bm{y}^{(k)}_{[g_{j}(i-1):g_{j}i]},\\hat{\\bm{y}}^{(k)}_{[g_{j}(i-1):g_{j}i]}),\n\\end{equation}\nwhere $M_{j}$ denotes the number of sliced segments. \nIn our case the set of $g_j$\\textquotesingle s was chosen as follows, $g_j \\in \\{4064, 2032, 1016, 508\\}$, assuming they moderately cover the range of duration of phonemes in speech.\n\nTo further improve the design of the loss function, we applied two simple techniques --- 1. pre-emphasis ($\\pi$) and 2. $\\mu$-law encoding ($\\mu$) --- on signals. \nAs most of the speech signal components are concentrated in the lower frequency bands, we found that applying pre-emphasis on loss function can help penalize high frequency components.\nIn addition, since the samples of speech signals are usually centered around zero, we found that it is helpful to use 16-bit $\\mu$-law encoding as it distributes samples more uniformly by the nature of continuous logarithmic transform.\nThe proposed loss function $\\mathcal{L}^{+}$ is defined as follows,\n\\begin{equation}\n\\begin{split}\n\\label{eq:emphasized}\n\t\\mathcal{L}^{+}(\\bm{y}^{(k)},\\hat{\\bm{y}}^{(k)}) &= \\mathcal{L}(\\bm{y}^{(k)},\\hat{\\bm{y}}^{(k)}) + \\mathcal{L}(\\pi(\\bm{y}^{(k)}),\\pi(\\hat{\\bm{y}}^{(k)})) \\\\\n\t& + \\mathcal{L}(\\mu(\\pi(\\bm{y}^{(k)})),\\mu(\\pi(\\hat{\\bm{y}}^{(k)})))\n\\end{split}\n\\end{equation}\nFinally, we used the proposed loss function for every $k$ and $\\lnot k$ combinations as follows, \n\\begin{equation}\n \\mathcal{L}_\\text{final} = \\sum_{k} ( \\mathcal{L}^{+}(\\bm{y}^{(k)},\\hat{\\bm{y}}^{(k)}) + \\mathcal{L}^{+}(\\bm{y}^{(\\lnot k)},\\hat{\\bm{y}}^{(\\lnot k)}) ).\n\\end{equation}\n\n\\section{Optimization for Real-Time U-Net}\n\\label{section:optimization}\nTo connect each encoder layer with its corresponding decoder, U-Net is often composed of convolutional layers with zero padding for dynamic input sizes. Without zero padding, the valid size of input and output is uniquely determined by the kernel sizes and strides. This obviously takes less computation and allows to keep only the essential part. In our real-time setting, the input spectrogram has 253 frequency bins and 65 frames by discarding the four lowest bins from the original spectrogram with a 512-point FFT, assuming that 16 kHz speech signals have no significant spectral component below 93.75 Hz. \nWe followed the network architecture of the real-valued U-Net proposed in \\cite{choi2019phase} (\\texttt{model10} and \\texttt{model20} specifically) with the modification of the last layer to output PHM. All the batch normalizations were fused into convolution filters.\n\nIn the encoder, the na\u00efve implementation of U-Net repeatedly performs the same computation that has already been computed previously.\nThis redundancy can be efficiently reduced by caching the pre-computed values using queues.\nLikewise, we utilized a similar concept with 2D convolution, but more than one queues are needed for the strided convolution in each layer.\nThe number of required queues for depth $d$ is derived by $\\prod_{l=1}^{d}{s_l}$, where $s_l$ denotes the temporal stride of the $l$-th encoder layer.\n\nMost computation of the na\u00efve U-Net is concentrated on a few decoder layers before the output. Fortunately, only a single frame of the output mask is needed for real-time inference. \nAlthough using the latest frame can achieve the shortest latency, it is better to preview a few milliseconds for performance.\nIt is computationally less efficient to use a longer lookahead because more frames should be calculated in the previous decoder layer. Our real-time implementation previews 32ms which is shorter than allowed in the DNS challenge \\cite{reddy2020interspeech}. The schematic details are shown in Fig. \\ref{fig:optimization}.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.47]{figs\/Model_optimization.pdf}\n\\caption{A graphical illustration of U-Net optimization for real-time inference. As a schematic view of 2D feature map, the number in the box indicates the relative index to the latest frame. $LA$ and $T$ denote the lookahead and the frame length respectively. The number of multiplications reduced from the na\u00efve one is shown at the bottom of the box (in millions). The overall reduction reached 88.9\\%.}\n\\label{fig:optimization}\n\\vspace*{-15pt}\n\\end{figure}\n\n\\section{Experiments}\n\\label{section:experiments}\n\\subsection{Dataset}\nWe used the DNS challenge dataset \\cite{reddy2020interspeech} and internally collected dataset for training.\nThe former is a large scale dataset where the speech samples were collected from Librivox \\cite{mcguirelibrivox}, and noise samples from Audioset \\cite{gemmeke2017audio} and Freesound \\cite{fonseca2017freesound}. Note that we did not use the provided noisy speech data from the DNS dataset but used on-the-fly augmentation with the clean speech and noise in the two datasets during the training phase.\nSince our goal is to perform both denoising and dereverberation, we used pyroomacoustics \\cite{scheibler2018pyroomacoustics} to simulate an artificial reverberation with randomly sampled absorption, room size, location of source and microphone distance.\nWe also trimmed random segments of 2 seconds from speech and noise data, and mixed them with uniformly distributed source-to-noise ratio (SNR) ranging from -10 dB to 30 dB.\n\nFor test, we used two datasets such as the synthesized testset in the DNS challenge (DNS) and WHAMR \\cite{maciejewski2019whamr}.\nThe DNS synthesized testset provides noisy-reverberant mixtures and noisy mixtures without reverb.\nDNS was used only to test the denoising performance since it does not provide the direct source signal of synthesized mixture samples. \nTherefore, a reverberant source was given as ground truth when the model is tested on noisy-reverberant mixtures.\nBoth the denoising and dereverberation performance were tested on the \\textit{min} subset of WHAMR dataset which contains 3,000 audio files.\nTo test the denoising and dereverberation performances both simultaneously and separately, we tested our models on four scenarios: \n1) \\textit{nr2d}: noisy-reverberant mixture to direct source 2) \\textit{nr2r}: noisy-reverberant mixture to reverberant source 3) \\textit{n2d}: noisy mixture to direct source 4) \\textit{r2d}: reverberant source to direct source.\nThe corresponding four pair of test subsets, denoted in a following way ($\\texttt{mixture}$, $\\texttt{ground\\_truth}$), were used as follows, 1. \\textit{nr2d}: ($\\texttt{mix\\_single\\_reverb}$, $\\texttt{s1\\_anechoic}$),\n2. \\textit{nr2r}: ($\\texttt{mix\\_single\\_reverb}$, $\\texttt{s1\\_reverb}$), 3. \\textit{n2d}: ($\\texttt{mix\\_single\\_anechoic}$, $\\texttt{s1\\_anechoic}$), 4. \\textit{r2d}: ($\\texttt{s1\\_reverb}$, $\\texttt{s1\\_anechoic}$).\n\n\\subsection{Implementation}\nInput features were used as a channel-wise concatenation of log-magnitude spectrogram, real and imaginary part of demodulated phase \\cite{takahashi2018phasenet}, group delay, and delta-phase \\cite{mccowan2011delta}.\nThe window size of \\texttt{model20} was 1024 with 256 hop size and the window size of \\texttt{model10} was 512 with 128 hop size.\nAll models were trained for 125k iterations with AdamW optimizer \\cite{reddi2019convergence}. The learning rate was set to 0.0004 and halved at 62.5k iteration.\nEvery test was done with a non-causal inference using \\texttt{model20} except the experiments in subsection \\ref{subsection:real-time}.\n\n\\subsection{Ablation studies}\n\\vspace{-10pt}\n\\begin{table}[ht]\n\\caption{The effect of proposed loss function. The \\textbf{denoising} performance was tested on the DNS challenge synthesized testset (\\textit{w\/o} and \\textit{w\/} reverb) and both \\textbf{denoising} and \\textbf{dereverberation} performance was tested on WHAMR dataset (\\textit{nr2d}: noisy-reverberant mixture to direct source, \\textit{r2d}: reverberant source to direct source).}\n\\begin{center}\n\\setlength\\tabcolsep{1.5pt}\n\\begin{tabular}{l|cc|cc|cc|cc}\n\\toprule\n\\multicolumn{9}{c}{DNS-challenge}\\\\\n\\midrule\n\\multicolumn{1}{c}{Loss}\n&\\multicolumn{2}{|c}{$\\mathbb{C}$MSE}&\\multicolumn{2}{|c}{SingleScale} &\\multicolumn{2}{|c}{MultiScale} &\\multicolumn{2}{|c}{MultiScale+}\n\\\\ \\midrule\n\\multicolumn{1}{c}{Reverb} &\\multicolumn{1}{|c}{\\textit{w\/o}} &\\multicolumn{1}{c}{\\textit{w\/}} &\\multicolumn{1}{|c}{\\textit{w\/o}}&\\multicolumn{1}{c}{\\textit{w\/}} &\\multicolumn{1}{|c}{\\textit{w\/o}}&\\multicolumn{1}{c}{\\textit{w\/}} &\\multicolumn{1}{|c}{\\textit{w\/o} }&\\multicolumn{1}{c}{\\textit{w\/}}\n\\\\ \\midrule\n\\bf{Si-SDR} & 15.63 & 14.21 & 17.47 & 15.79 & 17.57 & 15.93 & \\bf{17.91} & \\bf{16.22} \\\\\n\\bf{PESQ} & 2.22 & 2.59 & 2.57 & 2.90 & 2.63 & 2.97 & \\bf{2.71} & \\bf{3.01} \\\\\n\\toprule\n\\multicolumn{9}{c}{WHAMR}\\\\\n\\midrule\n\\multicolumn{1}{c}{Loss}\n&\\multicolumn{2}{|c}{$\\mathbb{C}$MSE}&\\multicolumn{2}{|c}{SingleScale} &\\multicolumn{2}{|c}{MultiScale} &\\multicolumn{2}{|c}{MultiScale+}\n\\\\ \\midrule\n\\multicolumn{1}{c}{Task} &\\multicolumn{1}{|c}{\\textit{nr2d}} &\\multicolumn{1}{c}{\\textit{r2d}} &\\multicolumn{1}{|c}{\\textit{nr2d}}&\\multicolumn{1}{c}{\\textit{r2d}} &\\multicolumn{1}{|c}{\\textit{nr2d}}&\\multicolumn{1}{c}{\\textit{r2d}} &\\multicolumn{1}{|c}{\\textit{nr2d} }&\\multicolumn{1}{c}{\\textit{r2d}}\n\\\\ \\midrule\n\\bf{Si-SDR} & 4.21 & 8.87 & 5.08 & 9.88 & 5.24 & 10.13 & \\bf{5.33} & \\bf{10.40} \\\\\n\\bf{PESQ} & 1.38 & 2.58 & 1.45 & 2.96 & \\bf{1.54} & 3.09 & 1.52 & \\bf{3.16} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{table:loss_comparison}\n\\end{table}\n\\vspace*{-15pt}\nTo show the effect of loss functions we observed SI-SDR \\cite{le2019sdr} and PESQ \\cite{rix2001perceptual} while changing four different loss functions. Complex MSE ($\\mathbb{C}$MSE) and three different cosine similarity based loss functions --- SingleScale, MultiScale, and MultiScale+ --- were compared each of which corresponds to Eq. \\ref{eq:cossim}, Eq. \\ref{eq:multiscale}, and Eq. \\ref{eq:emphasized}, respectively.\nThe quantitative results in Table \\ref{table:loss_comparison} show that the proposed multi-scale and emphasis functions are beneficial for both denoising and dereverberation tasks in most of the cases.\n\\subsection{Analysis on phase enhancement}\n\\begin{table}[ht]\n\\vspace*{-10pt}\n\\caption{\nPhase distance and gain under four different tasks.\n}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\toprule\nTask & \\textit{nr2r} & \\textit{n2d} & \\textit{nr2d} & \\textit{r2d} \\\\\n\\midrule\n$PD$($\\bm{Y}$, $\\bm{X}$) & 21.1\\textdegree & 23.3\\textdegree & 36.3\\textdegree & 24.7\\textdegree\\\\\n$PD$($\\bm{Y}$, $\\hat{\\bm{Y}}$) & 20.2\\textdegree & 21.9\\textdegree & 29.5\\textdegree & 15.0\\textdegree\\\\\n$PhaseGain$ & 4.5\\% &  6\\% & 17.6\\% & 64\\% \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{table:phasedist}\n\\vspace*{-25pt}\n\\end{table}\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[scale=0.4]{figs\/GD.png}\n\\caption{Group delay of enhanced phase}\n\\label{fig:group_delay}\n\\vspace*{-15pt}\n\\end{figure}\nHere, we used the phase distance defined in \\cite{choi2019phase} to quantitatively measure the phase enhancement performance.\nPhase distance ($PD$) between spectrogram $A$ and $B$ is formulated as follows,\n\\begin{equation}\n\\label{eq:phase_distance}\n  \\begin{aligned}\n\tPD(\\bm{A}, \\bm{B}) = \\sum_{t,f} \\frac{\\abs{A_{t,f}}}{\\sum_{t',f'}\\abs{A_{t',f'}}} \\angle(A_{t,f}, B_{t,f}),\n  \\end{aligned}\n\\end{equation}\nwhere $\\angle(A_{t,f}, B_{t,f})$ is the angle between $A_{t,f}$ and $B_{t,f}$, ranging from 0\\textdegree  to 180\\textdegree.\nWe measured $PD$ between ground truth $\\bm{Y}$ and mixture $\\bm{X}$, and $PD$ between ground truth $\\bm{Y}$ and estimation $\\bm{\\hat{Y}}$, and checked how much $PhaseGain(\\%)$ was obtained. This was tested on all four scenarios of WHAMR testset and shown in Table \\ref{table:phasedist}.\nWe found that the network is able to give a reasonable $PhaseGain$ in tasks including dereverberation (\\textit{nr2d}, \\textit{nr2r}).\nHowever, $PhaseGain$ was marginal for only-denoising-tasks (\\textit{nr2r}, \\textit{n2d}).\nWe conjecture that this is because the network is not able to estimate a precise magnitude value for noisy mixture and this issue is left for futurework.\nA visualization of enhanced phase group delay tested on a reverberant source is shown in Fig. \\ref{fig:group_delay}.\nFig. \\ref{fig:group_delay} (b) shows the enhanced harmonic structure of phase group delay.\n\n\\subsection{Computation of real-time U-Net}\n\\label{subsection:real-time}\n\\vspace*{-10pt}\n\\begin{table}[ht]\n\\caption{\nThe effect of causality and the model size.\n}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\toprule\nCausal\/Model & \\xmark\/NRT & \\cmark\/NRT & \\xmark\/RT & \\cmark\/RT \\\\\n\\midrule\n\\textbf{SI-SDR} & \\bf{5.33} & 4.60 & 3.42 & 2.33 \\\\\n\\textbf{PESQ} & \\bf{1.52} & 1.43 & 1.39 & 1.34 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{table:real-time}\n\\end{table}\n\\vspace*{-15pt}\nFollowing the constraint for real-time model suggested by the DNS challenge, we measured the elapsed time to compute a single frame. \\texttt{model20} that took 40 ms to compute a frame will be denoted as non-real-time (NRT) model, and \\texttt{model10} that took 4.32 ms to compute a frame will be denoted as real-time (RT) model. \nTo compare the two models and how causal inference affects the model performance, we compare four combinations in \\textit{nr2d} task. Table \\ref{table:real-time} shows that both non-causal inference and model size are significant factors for performance.\n\nFinally, we report the Mean Opinion Score (MOS) results from the DNS challenge based on the online subjective evaluation framework ITU-T P.808 \\cite{p808}. \nFor better perceptual quality, we linearly added the estimated direct source and reverberant source with a 15 dB ratio, and implemented a simple and zero-delay dynamic range compression to apply on it.\nOur causal-NRT and causal-RT model achieved a mean opinion score of 3.36 and 3.24, respectively.\n\n\\section{Conclusions}\nWe proposed a new mask and loss function to improve the performance of single-stage denoising and dereverberation.\nAs the proposed PHM and loss function are orthogonal to the network structure, we believe that a better performance can be achieved using the variant of U-Net architectures such as \\cite{takahashi2017multi, tolooshams2020channel}.\n\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and our results}\\label{sec1}\n\\subsection{Transversals and rainbow colourings} For $n\\in \\N$, let us write $[n]:=\\{1,\\dots, n\\}$. A {\\it Latin square} is an $n\\times n$ array filled with symbols from $[n]$, so that each symbol appears exactly once in each row and each column. A {\\it partial transversal} of a Latin square is a subset of its entries, each in a distinct row and column, and having distinct symbols. A partial transversal of size $n$ is a {\\it full transversal}.\n\nThe study of Latin squares goes back to Euler, who was, in particular, interested in finding Latin squares decomposable into full transversals. It is however not  obvious whether {\\it any} Latin square should have a large transversal. Ryser~\\cite{Ry67}, Stein~\\cite{St75} and Brualdi~\\cite{BR91} conjectured that any given Latin square has a partial transversal of size $n-1$ (it need not have a full one if $n$ is even). The current record towards this problem is due to Hatami and Shor \\cite{HS08}, who, correcting a mistake in an earlier work of Shor~\\cite{Sh82}, proved that there always exists a partial transversal of size $n-O(\\log^2 n)$.\n\nClearly, each symbol appears in a Latin square exactly $n$ times. A more general conjecture was made by Stein~\\cite{St75}, who suggested that any $n\\times n$ array filled with symbols from $[n]$, each appearing exactly $n$ times, has a partial transversal of size $n-1$. The best known positive result in this direction is due to Aharoni, Berger, Kotlar and Ziv \\cite{ABKZ17}, who, using a topological approach, showed that any such array has a partial transversal of size at least $2n\/3$.\nOn the other hand, Pokrovskiy and Sudakov \\cite{PS17a} recently disproved Stein's conjecture: in fact, they showed that there are such arrays with largest transversal of size $n-\\Omega(\\log n)$.\n\nEach $n\\times n$ array filled with symbols may be viewed as a colouring of a complete bipartite graph $K_{n,n}$: an edge $ij$ corresponds to the entry of the array in the $i$-th row and $j$-th column, and each symbol stands for a colour. In this way, a Latin square corresponds to a properly edge-coloured $K_{n,n}$, and a partial transversal is a {\\it rainbow matching} in $K_{n,n}$, that is, a collection of disjoint edges having pairwise distinct colours. Thus, the conjecture of Stein deals with (globally) $n$-bounded colourings of $K_{n,n}$, where we say that an edge-colouring of a graph is {\\it (globally) $g$-bounded} if each colour appears at most $g$ times in the colouring. An edge-colouring is {\\it locally $\\ell$-bounded} if each colour appears at most $\\ell$ times at any given vertex. Note that locally $1$-bounded colourings are simply proper colourings.\n\nStudying rainbow substructures in graphs has a long history. One source of inspiration is Ramsey theory, in particular, the canonical version of Ramsey's theorem due to Erd\\H os and Rado \\cite{ER50}. A general problem is to find conditions on the colourings and graphs which would allow to find certain rainbow substructures. This topic has received considerable attention recently, with probabilistic tools and techniques from extremal graph theory allowing for major progress on longstanding problems. In this context, natural (rainbow) structures to seek include matchings, Hamilton cycles, spanning trees and triangle factors (see e.g.~\\cite{AFR95,APS17,CP17, CKPY,GRWW, GJ18,MPS18,P16, PS17b}).  It is easy to see that results on edge-coloured $K_n$ also imply results on patterns in symmetric $n\\times  n$ arrays.\n\n\\subsection{(Almost) spanning rainbow structures in complete graphs} Andersen~\\cite{An89} conjectured that every properly edge-coloured $K_n$ contains a rainbow path of length $n-2$ (which would be best possible by a construction of\nMaamoun and Meyniel~\\cite{MM84}).\nDespite considerable research, even the existence of an almost\nspanning path or cycle was a major open question until recently.\nAlon, Pokrovskiy and Sudakov \\cite{APS17} were able to settle this\nby showing that any properly edge-coloured $K_n$ contains a rainbow cycle of length $n-O(n^{3\/4})$ (the error term was subsequently improved in \\cite{BM17}). A corollary of our second main theorem (Theorem~\\ref{thm: near spanning cycle}) states that we can arrive at a stronger conclusion (i.e.~we obtain many edge-disjoint\nalmost-spanning rainbow cycles) under much weaker assumptions (though with a larger error term).\nNote that, similarly to the case of Latin squares, any proper edge-colouring of $K_n$ is $n\/2$-bounded.\n\\begin{corollary}\\label{cor1}\n  Any  $(1+o(1))n\/2$-bounded, locally $o(n)$-bounded edge-colouring of $K_n$ contains $(1-o(1))n\/2$ edge-disjoint rainbow cycles of length $(1-o(1))n$.\n\\end{corollary}\nAs noted above, even for proper colourings, the corollary is best possible up to the value of the final error term, i.e.~we cannot guarantee a Hamilton cycle.\nMoreover, a slight modification of the construction of Pokrovskiy and Sudakov in~\\cite{PS17a}, shows that there are locally $o(n)$-bounded, $(n-1)\/2$-bounded edge-colourings of $K_n$ with no rainbow cycle longer than $n-\\Omega(\\log n)$.\nFor a more detailed discussion, see Section~\\ref{concl}.\n\n\nIt is, however, more desirable to have spanning (rather than almost-spanning) structures. Which conditions guarantee the existence of a rainbow Hamilton cycle? Albert, Frieze and Reed \\cite{AFR95} showed that there exists $\\mu >0$, such that in any $\\mu n$-bounded edge-colouring of $K_n$ there is a rainbow Hamilton cycle. Their result was greatly extended by B\\\"ottcher, Kohayakawa, and  Procacci~\\cite{BKP}, who showed that any $n\/(51\\Delta^2)$-bounded edge-colouring of $K_n$ contains a rainbow copy of $H$ for any $n$-vertex graph $H$ with maximum degree at most $\\Delta$.\n\nNote that these requirements are quite strong compared to the trivial (global) $(n-1)\/2$-boundedness condition which is the limit of what one could hope for. If we impose a global bound of $(1-o(1))n\/2$ on the sizes of each colour class, then it turns out that we can still guarantee rainbow spanning structures, provided some moderate local boundedness conditions hold. The following is a corollary of  our third and fourth main theorems (see Theorems~\\ref{thm: perfect decomp} and~\\ref{thm: spanning cycle}). For  given graphs $F$ and $G$, we say that $L \\subseteq G$ is an {\\it $F$-factor} if $L$ consists of vertex-disjoint copies of $F$ covering all vertices of $G$.\n\n\\begin{corollary}\\label{cor2}\n  For any $\\eps>0$, there exist $\\eta>0$ and $n_0$ such that for all $n\\ge n_0$, any $(1-\\eps)\\frac{n}2$-bounded, locally $\\frac{\\eta n}{\\log^{4} n}$-bounded edge-colouring of $K_n$ contains a rainbow Hamilton cycle and a rainbow triangle-factor (assuming that $n$ is divisible by $3$ in the latter case).\n\\end{corollary}\nIn particular, any proper, $(1-o(1))n\/2$-bounded edge-colouring of $K_n$ contains a rainbow Hamilton cycle. Bipartite versions of this, where one of the aims  is to find rainbow perfect matchings in $(1-o(1))n$-bounded edge-colourings of $K_{n,n}$,\nhave been intensively studied, see e.g.~\\cite{HJ08,P15}.\n\nCorollary~\\ref{cor2} is best possible in the following sense:\n as mentioned above, a proper (and thus $n\/2$-bounded) edge-colouring of $K_n$ does not guarantee a rainbow Hamilton cycle. In fact, this condition does not even ensure the existence of $n$ different colours required for a Hamilton cycle.\n\n\n\\subsection{(Approximate) decompositions of complete graphs into rainbow structures} As already mentioned, Euler was interested in finding Latin squares that are decomposable into full transversals.\nThis corresponds to finding decompositions of properly edge-coloured complete bipartite graphs\n$K_{n,n}$ into perfect rainbow matchings. More generally, we say that a graph $G$ has a {\\it decomposition} into graphs $H_1,\\ldots, H_k$ if $E(G)=\\bigcup_{i=1}^k E(H_i)$ and the edge sets of the $H_i$ are pairwise disjoint. The existence of various decompositions of $K_n$ is a classical topic in design theory, related to Room squares \\cite{W74}, Howell designs \\cite{R78} and Kotzig factorizations \\cite{CM82}. In the setting of these questions, however, one is allowed to construct both the colouring and the decomposition. But, once again, it is natural to ask what one can say for arbitrary colourings with certain restrictions.\n\nThe most studied case is that of decompositions into trees. The following conjecture was raised, with some variations, by Brualdi and Hollingsworth \\cite{BH96}, Kaneko, Kano, and Suzuki \\cite{KKS02} and Constantine \\cite{C02}: prove that every properly coloured complete graph is (almost) decomposable into (possibly isomorphic) rainbow spanning trees. Recently Pokrovskiy and Sudakov \\cite{PS17b} as well as Balogh, Liu and Montgomery \\cite{BLM17} independently showed that in a properly edge-coloured $K_n$ one can find a collection of linearly many edge-disjoint rainbow spanning trees.\n\nOur results actually work in the setting of {\\it approximate decompositions}.  We say that a collection of edge-disjoint subgraphs $L_1,\\ldots, L_t$ of $G$ is an {\\it $\\eps$-decomposition of $G$}, if they contain all but at most an $\\eps$-proportion of the edges of $G$. The following result is a special case of Theorem~\\ref{thm: spanning cycle}.\n\n\\begin{corollary}\\label{cor3}\n  For any $\\eps>0$, there exist $\\eta>0$ and $n_0$ such that for all $n\\ge n_0$,  any $(1-\\eps)\\frac{n}2$-bounded, locally $\\frac{\\eta n}{\\log^{4} n}$-bounded edge-colouring of $K_n$ has an $\\eps$-decomposition into rainbow Hamilton cycles.\n\\end{corollary}\n\nNote that this corollary implies an approximate version of the three conjectures on decompositions into spanning rainbow trees mentioned above. Indeed, for proper edge-colourings of $K_n$ with an additional mild restriction on the size of each colour class ($(1-\\eps)n\/2$ instead of $n\/2$), rainbow Hamilton cycles with one edge removed give us an approximate decomposition into isomorphic spanning paths. Similarly, Corollary~\\ref{cor1} also implies an approximate version of the above conjectures as it gives (without any restriction on the sizes of the colour classes) an approximate decomposition into almost-spanning paths.\n\n\n\\subsection{Rainbow spanning structures and decompositions in quasirandom graphs}\\label{sec2} Our results\nactually hold not only for colourings of $K_n$, but in the much more general setting of quasirandom graphs (and thus for example with high probability for dense random graphs). One of our main proof ingredients is a recent powerful result of Glock and Joos \\cite{GJ18}, who proved a rainbow blow-up lemma which allows to find rainbow copies of spanning subgraphs in a suitably quasirandom graph $G$, provided that the colouring is $o(n)$-bounded (see Theorem~\\ref{blowup}). As a consequence, they proved a rainbow bandwidth theorem under the same condition on the colouring. Note however that their blow-up lemma does not directly apply in our setting, as the restriction on the colouring is much stronger than in our case. We nevertheless can use it in our proofs since we apply it in a small random subgraph, on which the colouring has the necessary boundedness condition.\n\n\nTo formulate our results, we need the definition of a quasirandom graph. This will require some preparation. For $a,b,c\\in \\mathbb{R}$ we write $a = b\\pm c$ if $b-c \\leq a \\leq b+c$. We define $\\binom{X}{k}:=\\{A\\subseteq X: |A|=k\\}$.\nFor a vertex $v$ in a graph $G$, let $d_G(v)$ denote its degree and $N_G(v)$ its set of neighbours. The maximum and minimum degrees of $G$ are denoted by $\\Delta(G)$ and $\\delta(G)$, respectively. For $u,v\\in V(G)$, we put $N_G(u,v) := N_G(u)\\cap N_G(v)$ and $d_{G}(u,v)=|N_{G}(u,v)|$. The latter function we call the {\\it codegree} of $u$ and $v$. We will sometimes omit the subscript $G$ when the graph is clear from the context.\n\nWe say that an $n$-vertex graph $G$ is {\\it $(\\epsilon,d)$-quasirandom} if\n$d(v)= (d\\pm \\epsilon)n$ for each $v\\in V(G)$ and\n\\begin{equation}\\label{eq: irreg} \\Big|\\big\\{uv \\in \\binom{V(G)}{2}: d(u,v)\\neq (d^2\\pm \\epsilon)n\\big\\}\\Big|\\le \\eps n^2.\\end{equation}\n\n\nNote that this is weaker than the standard notion of $(\\eps, d)$-quasirandomness, where the set of exceptional vertex pairs having the ``wrong'' codegree is required to be empty (on the other hand, our notion is very close to the classical notion of\n$\\eps$-superregularity). Our first  theorem guarantees the existence of an approximate decomposition into almost-spanning $F$-factors.\nFor graphs $F$, $G$ and $0\\le \\alpha\\le 1$, we say that $L$ is an {\\it $\\alpha$-spanning $F$-factor in $G$}, if $L$ is a subgraph of $G$, consisting of vertex-disjoint copies of $F$ and containing all but at most an $\\alpha$-proportion of the vertices of $G$. We define an {\\it $\\alpha$-spanning cycle in $G$} analogously.\n\n\\begin{theorem}\\label{thm: approx decomp}\nFor given $\\alpha,d_0>0$ and $f,h\\in \\mathbb N$, there exist $\\eta>0$ and $n_0$ such that the following holds for all $n\\ge n_0$ and $d\\ge d_0$. Suppose that $G$ is an $n$-vertex $(\\eta,d)$-quasirandom graph and $F$ is an $f$-vertex $h$-edge graph. If $\\phi$ is a $(1+\\eta)\\frac{f dn}{2h}$-bounded, locally  $\\eta n$-bounded  edge-colouring of $G$, then $G$ contains an $\\alpha$-decomposition into rainbow $\\alpha$-spanning $F$-factors.\n\\end{theorem}\nNote that the $(1+o(1))\\frac{f dn}{2h}$-boundedness of the colouring cannot be replaced by a weaker condition even for a single $o(1)$-spanning $F$-factor, since we are only guaranteed roughly $|E(G)|\/((1+o(1))\\frac{f dn}{2h})=(1-o(1))\\frac{hn}{f}$ distinct colours in such a colouring. On the other hand, an $o(1)$-spanning $F$-factor also contains $(1-o(1))\\frac{hn}{f}$ edges of distinct colours. In the case when $F$ is an edge (i.e. when we are looking for an almost perfect rainbow matching), a much stronger conclusion holds:\nwe can in fact drop the quasirandomness condition and consider much sparser graphs (see Section~\\ref{concl}).\n\n\nThe next theorem guarantees the existence of an approximate decomposition into almost-spanning rainbow cycles.\n\n\\begin{theorem}\\label{thm: near spanning cycle}\nFor given $\\alpha,d_0>0$, there exist $\\eta>0$ and $n_0$ such that the following holds for all $n\\ge n_0$ and $d\\ge d_0$.\nSuppose that $G$ is an $n$-vertex $(\\eta,d)$-quasirandom graph.\nIf $\\phi$ is a $\\frac{1}{2}(1+\\eta)dn$-bounded, locally $\\eta n$-bounded edge-colouring of $G$, then $G$ contains an $\\alpha$-decomposition into rainbow $\\alpha$-spanning cycles.\n\\end{theorem}\n\nFor the same reasons as in Theorem~\\ref{thm: approx decomp}, the $\\frac{1}{2}(1+\\eta)dn$-boundedness condition cannot be replaced by a significantly weaker one.\n\n\nIf we slightly strengthen both the local and the global boundedness condition, we can obtain spanning structures, as guaranteed by the next two theorems below.\nThe first theorem guarantees the existence of an approximate decomposition into rainbow $F$-factors. Let us denote $a(F):=\\max\\{\\Delta(F),a'(F),a''(F)\\},$ where $a'(F)$ is the maximum of the expression $d(u)+d(v)-2$ over all edges $uv\\in E(F)$, and $a''(F)$ is the maximum of the expression $d(u)+d(v)+d(w)-4$ over all paths $uvw$ in $F$. Note that $a(F)\\le \\max\\{\\Delta(F),3\\Delta(F)-4\\}$.\n\n\n\\begin{theorem} \\label{thm: perfect decomp}\nFor given $\\alpha,d_0>0$  and $a,f,h\\in \\mathbb N$, there exist $\\eta>0$ and $n_0$ such that the following holds for all $n\\ge n_0$ which are divisible by $f$ and all $d\\ge d_0$. Suppose that $F$ is an $f$-vertex $h$-edge graph with $a(F)\\le a$.\nSuppose that $G$ is an $n$-vertex $(\\eta,d)$-quasirandom graph.\nIf $\\phi$ is a $(1-\\alpha) \\frac{fdn}{2h}$-bounded, locally $\\frac{\\eta n}{ \\log^{2a}{n}}$-bounded edge-colouring of $G$, then $G$ has an $\\alpha$-decomposition into rainbow $F$-factors.\n\\end{theorem}\n\n\nIn a similar setting, we can also obtain an approximate decomposition into rainbow spanning cycles.\n\n\n\\begin{theorem}\\label{thm: spanning cycle}\nFor given $\\alpha,d_0>0$, there exist $\\eta>0$ and $n_0$ such that the following holds for all $n\\ge n_0$ and $d\\ge d_0$.\nSuppose that $G$ is an $n$-vertex $(\\eta,d)$-quasirandom graph.\nIf $\\phi$ is a $\\frac{1}{2}(1-\\alpha)dn$-bounded, locally  $\\frac{\\eta n}{\\log^4{n}}$-bounded  edge-colouring of $G$, then $G$ contains an $\\alpha$-decomposition into rainbow Hamilton cycles.\n\\end{theorem}\nWe will discuss multipartite analogues of our results in Section~\\ref{concl}. (Recall that the bipartite case is of particular interest, as such results can be translated into the setting of arrays.)\nThere are numerous open problems that arise from the above results: in particular, it is natural to seek decompositions into more general rainbow structures such as regular spanning graphs of bounded degree. It would also be very desirable to obtain improved error terms or even exact results.\n\nThe remainder of this paper is organized as follows. In Section~\\ref{sec3} we collect the necessary definitions and auxiliary results, some of which are new and may be of independent interest (in particular, we prove a result on matchings in not\nnecessarily regular hypergraphs). In Section~\\ref{sec5}, we prove Theorems~\\ref{thm: approx decomp} and~\\ref{thm: near spanning cycle}. In Section~\\ref{sec6} we prove Theorems~\\ref{thm: perfect decomp} and~\\ref{thm: spanning cycle}. In Section~\\ref{concl}, we add some concluding remarks. In the appendix we prove the rainbow counting lemma, which plays an important role in the proofs.\n\n\n\n\n\\section{Preliminaries}\\label{sec3}\nIn this section, we introduce and derive several key tools that we will need later on: in particular, we state the rainbow blow-up lemma from \\cite{GJ18} and derive a result on random matchings in (not necessarily regular) hypergraphs as well as two probabilistic partition results.\n\n\n\n\\subsection{Notation}\nIn order to simplify the presentation, we omit floors and ceilings and treat large numbers as integers whenever this does not affect the argument. The constants in the hierarchies used to state our results have to be chosen from right to left. More precisely, if we claim that a result holds whenever $1\/n \\ll a \\ll b \\leq 1$ (where $n\\in \\N$ is typically the order of a graph), then this means that there are non-decreasing functions $f^* : (0, 1] \\rightarrow (0, 1]$ and $g^* : (0, 1] \\rightarrow (0, 1]$ such that the result holds for all $0 < a, b \\leq 1 $ and all $n \\in \\mathbb{N}$ with $a \\leq f^*(b)$ and $1\/n \\leq g^*(a)$. We will not calculate these functions explicitly.\n\nThe auxiliary hierarchy constants used in this paper will be denoted by the Greek letters from $\\alpha$ to $\\eta$ (reserved throughout for this purpose).\nIn what follows, $n$ is the number of vertices in a graph or a part of a multipartite graph;\n$d$  stands for the density of a graph.\nWe use $i,j,k$, along with possible primes and subscripts, to index objects. We use letters $u,v,w$ to denote vertices and $e$ to denote graph edges. Colours are usually denoted by $c$ and the colouring itself by $\\phi$, while capital $C$  (with possible subscripts) stands for various constants. We reserve other capital Latin letters except $N$ for different sets or graphs. In the case of graphs or sets, having a prime in the notation means that later in the proof\/statement we refine this object by removing some exceptional elements (note that primes do not have this meaning for the indexing variables). Of course, a double prime will then mean that we remove the exceptional elements in two stages. Calligraphic letters will stand for collections of sets, such as partitions or hypergraphs.\n\n\nAll graphs considered in this paper are simple. However, we allow our hypergraphs to have multiple edges. We use standard notations $V(\\cdot)$ and $E(\\cdot)$ for  vertex and edge sets of graphs and hypergraphs. The number of edges in a graph $G$ is denoted by $e(G)$. For a vertex set $U$ and an edge set $E$, we denote by $G\\setminus U$ the graph we obtain from $G$ by deleting all vertices in $U$ and $G-E$ denotes the graph we obtain from $G$ by deleting all edges in $E$.  For a set $U\\subseteq V(G)$ and $u,v\\in V(G)$, we put\n\\begin{align*}\nd_{G,U}(u):= |N_{G}(u)\\cap U| \\ \\ \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\\n\\ d_{G,U}(u,v) &:= |N_{G}(u,v)\\cap U|.\n\\end{align*}\nFor a graph $G$ and two disjoint sets $U,V\\subseteq V(G)$, let\n$G[U,V]$ denote  the graph with  vertex set $U\\cup V$ and  edge set $\\{uv \\in E(G) : u\\in U, v\\in V\\}$.\nMore generally, given disjoint sets $U_1,\\ldots, U_k\\subseteq V(G)$, we define the $k$-partite subgraph $G[U_1,\\ldots, U_k]$ of $G$ in a similar way.\nWe denote by $P_k$ a path with $k$ edges.\n\nSince in this paper we deal with edge-colourings only, we simply refer to them as {\\it colourings}. For shorthand, we call a colouring $\\phi: E(G)\\rightarrow [m]$ of $G$ in $m$ colours an \\emph{$m$-colouring} of $G$. We denote by $G(\\phi,c)$ the spanning subgraph of $G$ that contains all its edges of colour $c$ in $\\phi$.\nMore generally, for a set $I\\subseteq [m]$, we put $G(\\phi,I)=\\bigcup_{c\\in I} G(\\phi,c)$. An $m$-colouring $\\phi$ is {\\it $g$-bounded} if and only if $e(G(\\phi,c))\\leq g$ for each $c\\in [m]$ and  is \\emph{locally $\\ell$-bounded} if and only if $\\Delta(G(\\phi,c))\\leq \\ell$ for each $c\\in [m]$. We say that $\\phi$ is \\emph{$(g,\\ell)$-bounded} if it is $g$-bounded and locally $\\ell$-bounded.\n\n\n\n\n\n\n\\subsection{Probabilistic tools}\nIn this section, we collect the large deviation results we need.\n\\begin{lemma}[Chernoff-Hoeffding's inequality, see \\cite{JLR00}] \\label{Chernoff}\nSuppose that $X_1,\\dots, X_N$ are independent random variables taking values $0$ or $1$. Let $X=\\sum_{i\\in [N]} X_i$.\nThen $$\\mathbb{P}[|X - \\mathbb{E}[X]| \\geq t] \\leq 2e^{-\\frac{t^2}{2(\\mathbb{E}[X]+t\/3)}}.$$\n\\end{lemma}\nIn particular, if $t\\ge 7\\mathbb \\EXP[X]$, then $\\mathbb{P}[|X - \\mathbb{E}[X]| \\geq t] \\leq 2e^{-t}$.\n\nWe shall need two large deviation results for martingales.\n\\begin{theorem}[Azuma's inequality \\cite{Azu67}]\\label{Azuma}\nSuppose that $\\lambda>0$ and let $X_0,\\dots, X_N$ be a martingale such that\n$|X_{i}- X_{i-1}|\\leq \\vartheta_i$ for all $i\\in [N]$.\nThen\n\\begin{align*}\n\\mathbb{P}[\\left|X_N-X_0\\right|\\geq \\lambda]\\leq 2e^{\\frac{-\\lambda^2}{2\\sum_{i\\in [N]}\\vartheta_i^2}}.\n\\end{align*}\n\\end{theorem}\n\\begin{theorem}[\\cite{CK}, Theorems 6.1 and 6.5] \\label{Azuma2}\nSuppose that $\\lambda>0$ and let $X_0,\\dots, X_N$ be a martingale such that\n$|X_{i}- X_{i-1}|\\leq \\vartheta$ and $\\mathrm{Var}[X_i\\mid X_0,\\ldots,X_{i-1}]\\le \\sigma_i^2$ for all $i\\in [N]$.\nThen\n\\begin{align*}\n\\mathbb{P}[\\left|X_N-X_0\\right|\\geq \\lambda]\\leq 2e^{\\frac{-\\lambda^2}{2\\sum_{i\\in [N]}\\sigma_i^2+\\lambda \\vartheta}}.\n\\end{align*}\n\\end{theorem}\n\n\n\n\n\n\n\\subsection{Regularity} In this part, we discuss the relation between quasirandomness and superregularity, as well as collect some tools to deal with ``exceptional'' pairs of vertices that have high codegree.\n\nWe say that a bipartite graph $G$ with parts $U,V$ is \\emph{$(\\epsilon,d)$-regular} if for all sets $X\\subseteq U$, $Y\\subseteq V$ with  $|X|\\geq \\epsilon |U|$, $|Y|\\geq \\epsilon |V|$ we have\n\\begin{align*}\n\t\\Big| \\frac{e(G[X,Y])}{|X||Y|} - d \\Big| \\leq \\epsilon.\n\t\\end{align*}\nIf $G$ is $(\\epsilon,d)$-regular and $d_{G}(u)= (d \\pm \\epsilon)|V|$ for all $u\\in U$, $d_{G}(v) = (d \\pm \\epsilon)|U|$ for all $v\\in V$, then we say that $G$ is {\\em $(\\epsilon,d)$-superregular}. We remark that, although the notions of $\\eps$-superregularity and  $(\\eps,d)$-quasirandomness imply very similar properties of graphs, it is much handier to use the first one for bipartite graphs and the second one for general graphs. The following observation follows directly from the definitions, so we omit the proof.\n\n\n\n\\begin{proposition}\\label{prop: edge deletion regular}\nSuppose $1\/n \\ll \\epsilon\\ll \\delta \\ll d \\leq 1$. \\begin{itemize}\\item[(i)]\nIf $G'$ is an $(\\epsilon,d)$-regular bipartite graph with vertex partition $V_1, V_2$ with $|V_1|, |V_2|\\geq n$\nand $E\\subseteq E(G')$ is a set of edges with $|E| \\leq \\epsilon n^2$, then $G'-E$ is $(\\delta,d)$-regular.\n\\item[(ii)] Suppose $G'$ is an $(\\epsilon,d)$-quasirandom $n$-vertex graph, $E\\subseteq E(G')$ is a set of edges with $|E| \\leq \\epsilon n^2$ and $V\\subseteq V(G')$ is a set of vertices with $|V|\\le \\eps n$. Then $(G'\\setminus V)-E$ contains a $(\\delta,d)$-quasirandom subgraph $G$ on at least $(1-\\delta)n$ vertices.\n \\end{itemize}\n\\end{proposition}\n\n\n\nIn quasirandom graphs defined as in \\eqref{eq: irreg} there are exceptional pairs of vertices that have ``incorrect'' codegree. Similarly, in locally bounded colourings some pairs of vertices have large ``monochromatic codegree''. To deal with such exceptional pairs of vertices we introduce {\\it irregularity graphs}.\n\nFor an $n$-vertex graph $G$, we define the {\\it irregularity graph} $\\Ir_G(\\epsilon,d)$ to be the graph on $V(G)$ and whose edge set is as defined in \\eqref{eq: irreg}, i.e. $uv\\in E(\\Ir_G(\\eps,d))$ if and only if $d_G(u,v)\\ne (d^2\\pm \\eps) n$. Similarly, for a partition $\\cV$ of $V(G)$ into subsets $V_1,\\dots,V_r$, we let\n$\\Ir_{G,\\cV}(\\epsilon,d)$ be the graph on $V(G)$ with the edge set\n$$\\Big\\{uv \\in \\binom{V(G)}{2}: u\\in V_j, v\\in V_{j'},  d_{G}(u,v)\\neq (d^2\\pm \\epsilon)|V_{j''}| \\text{ for some } j''\\in [r]\\setminus\\{j,j'\\}\\Big\\}.$$ (Here we allow $j=j'$.)\n\n\\begin{theorem}\\cite{DLR95} \\label{thm: almost quasirandom}\nSuppose $0<1\/n\\ll\\epsilon \\ll \\alpha,d \\leq 1$.\nSuppose that $G$ is a bipartite graph with a vertex partition $\\cV=(U,V)$ such that $n=|U| \\leq |V|$.\nIf $e(\\Ir_{G,\\cV}(\\epsilon,d))\\leq \\epsilon n^2$ and $d(u)=(d\\pm \\eps)|V|$ for all but at most $\\eps n$ vertices $u\\in U$, then $G$ is $(\\epsilon^{1\/6},d)$-regular.\\COMMENT{This is weaker than the result in \\cite{DLR95} since they only need that almost all pairs in $U$ have the right codegree (e.t. they do not need anything for the pairs in $V$.}\n\\end{theorem}\n\nThe following lemma is an easy consequence of Theorem~\\ref{thm: almost quasirandom} and the definition of $\\epsilon$-superregularity. Thus we omit the proof.\n\\begin{lemma}\\label{lem: irregular degree}\nSuppose $0<1\/n\\ll\\epsilon \\ll 1\/r, \\alpha,d \\leq 1$.\n\\begin{itemize}\\item[(i)] Suppose that $G$ is an $n$-vertex, $(\\epsilon,d)$-quasirandom graph. Then\n$\\Delta(\\Ir_{G}(\\epsilon^{1\/10},d))\\leq  \\epsilon^{1\/10} n$.\n\\item[(ii)] Suppose that $\\cV=(V_1,\\dots, V_r)$ is a partition of $G$ such that $n\\leq |V_i| \\leq \\alpha^{-1}n$ for each $i\\in[r]$ and $G[V_i,V_j]$ is $(\\epsilon,d)$-superregular for all $i\\neq j\\in [r]$. Then $\\Delta(\\Ir_{G,\\cV}(\\epsilon^{1\/10},d))\\leq \\epsilon^{1\/10} n$.\n    \\end{itemize}\n\\end{lemma}\\COMMENT{\nTo prove (ii), for each $u\\in V(G)$, we take a neighbourhood of $u$. Then only small portion of vertices have wrong number of neighbours in the neighbourhood of $u$.\nThis shows that $u$ has low degree in $\\Ir_{G,\\cV}(\\epsilon^{1\/10},d)$.\nFor the quasirandom case (i), we consider two duplicates of $V(G)$, and consider a bipartite graph $G'$ between duplicates, such that $N_{G'}(u)$ is exactly the duplicates of $N_{G}(u)$ on the other side. Then Theorem~\\ref{thm: almost quasirandom} shows that $G'$ is $(2\\epsilon^{1\/6},d)$-regular. Now apply the argument for (ii) to $G'$.\n}\n\nFor $u,v\\in V(G)$ and a colouring $\\phi$ of $G$, let $C_{G}^{\\phi}(u,v):= \\{ w\\in N_{G}(u,v) : \\phi(uw)=\\phi(vw)\\}$ and let $c_{G}^{\\phi}(u,v)$ be its size, that is, the {\\it monochromatic codegree of $u,v$}.\n\nFor a given colouring $\\phi$ of $G$, we define the \\emph{colour-irregularity} graph\n$\\Ir_{G}^{\\phi}(\\ell)$ to be the graph on vertex set $V(G)$ and edge set\n$\\{ uv \\in \\binom{V(G)}{2} : c^{\\phi}_{G}(u,v) \\geq \\ell\\}$.\\COMMENT{It may contain a pair which is not an edge of $G$.}\nIn  words, we include a pair $uv$ in the edge set if there are at least $\\ell$ choices of $w \\in N_{G}(u,v)$ such that $\\phi(uw)=\\phi(vw)$.\n\\COMMENT{\nThe fact that $\\Ir_{G}(\\epsilon,d)\\cup \\Ir_{G}^{\\phi}(\\epsilon n)$ has maximum degree $o(n)$ is the fundamental reason why we can obtain our results.\n}\n\n\n\n\\begin{lemma}\\label{lem: colour irregular degree}\nLet $\\ell, n\\in \\N$.\nIf $\\phi$ is a locally $\\ell$-bounded colouring of an $n$-vertex graph $G$, then we have $\\Delta(\\Ir_{G}^{\\phi}(\\sqrt{\\ell n})) \\leq \\sqrt{\\ell n}$.\n\\end{lemma}\n\\begin{proof}\nSuppose that for some vertex $v$ there is a set $U$ of more than $\\sqrt{\\ell n}$ vertices $u$ such that $c^{\\phi}_G(u,v)\\geq \\sqrt{\\ell n}$.\nFor each $u\\in U$, consider the set $C^{\\phi}_G(u,v) \\subseteq N_G(v)$, which is of size at least $\\sqrt{\\ell n}$. In total, we have more than $\\sqrt{\\ell n}$ such sets of size $\\sqrt{\\ell n}$, and thus there exists a vertex $w\\in N_G(v)$ which belongs to $C^{\\phi}_G(u,v)$ for more than  $\\ell n\/d_{G}(v)\\geq \\ell$ vertices $u\\in U$.\nTake some $\\ell+1$ of these vertices, say, $u_1\\ldots, u_{\\ell +1}$.  We have $\\phi(u_iw)= \\phi(u_jw)$ for all $i,j\\in[\\ell+1]$, which contradicts the assumption that $\\phi$ is locally $\\ell$-bounded.\n\\end{proof}\n\n\\subsection{Counting rainbow subgraphs}\n\n\nIn the proof of Theorems~\\ref{thm: approx decomp} and~\\ref{thm: perfect decomp}, we deal with rainbow $F$-factors. The proofs of these theorems rely on a hypergraph-matching result in the spirit of the R\\\"odl nibble and the Pippenger-Spencer theorem (Theorem~\\ref{lem: Pippenger} below). To make the transition from hypergraphs to coloured graphs, roughly speaking, we associate a hyperedge with each rainbow copy of $F$. We will need to ensure that the degree and codegree conditions hold for the auxiliary hypergraph in order for the nibble machinery to work. Therefore, we need certain results that will allow us to estimate the number of rainbow copies of $F$ in a quasirandom (or superregular) graph~$G$.\n\nFor given graphs $F,G$, a subgraph $H$ of $G$ and a colouring $\\phi$ of $G$, we denote by $R_G^\\phi(F,H)$  the collection of $\\phi$-rainbow subgraphs $\\bar F$ of $G$ that are isomorphic\\COMMENT{$\\bar F\\simeq F$.} to $F$ and contain $H$ as an induced subgraph.\\COMMENT{$H\\subseteq \\bar F$. Note that $\\bar F, H$ are subgraphs of $G$ while $F$ is a graph disjoint from $G$.} Normally, $\\phi$ is obvious from the context, so we often omit it from the notation.\n\n\nFor a vertex partition $\\cX=\\{X_1,\\dots, X_{r'}\\}$\\COMMENT{Not necessarily a proper colouring} of $F$ and a collection $\\cV=\\{V_1,\\dots, V_r\\}$ of disjoint subsets of $V(G)$, we say that an embedding $\\psi$ of $F$ into $G$ or a copy $\\psi(F)$ of $F$ in $G$ \\emph{respects} $(\\cX,\\cV)$, if there exists a injective map $\\pi:[r']\\rightarrow [r]$ such that $\\psi(X_i) \\subseteq V_{\\pi(i)}$ for each $i\\in [r']$. By abuse of notation, we also use $V(F)$ to denote the partition of the vertex set of $F$ into singletons.\n\nFor a subgraph $H\\subseteq G$\nwe denote by $R_{G,\\cX,\\cV}(F,H)$ the collection of $\\phi$-rainbow copies $\\bar F$ of $F$ in $G$ that respect $(\\cX,\\cV)$ and that contain $H$ as an induced subgraph.\\COMMENT{We mainly use this definition for the case when $H$ is $K_1$ or $K_2$. In the former case, it's counting the rainbow copies containing a specific vertex. In the latter case, it's counting the rainbow copies containing a specific edge.}\nPut $r_{G,\\cX,\\cV}(F,H) := |R_{G,\\cX,\\cV}(F,H)|$. If $H$ is the order-zero graph\\COMMENT{that is a subgraph of every graph}, then we omit $H$ from the expression.\nIf $H$ is a vertex $v\\in V(G)$ or an edge $uv\\in E(G)$, then we write $R_{G,\\cX,\\cV}(F,v)$ or $R_{G,\\cX,\\cV}(F,uv)$, respectively.\nNote that $R_{G,\\cX,\\cV}(F,uv)$ and $R_{G,\\cX,\\cV}(F,\\{u,v\\})$ are different since the former does not count the rainbow copies of $F$ containing $u,v$ but not the edge $uv$, while the latter counts only those.\n\nFor a given graph $F$ with a vertex partition $\\cX=\\{X_1,\\dots, X_r\\}$ of $V(F)$ into independent sets, let $\\Aut_{\\cX}(F)$ denote the set of automorphisms $\\pi$ of $F$ such that $\\{X_1,\\dots, X_r\\} = \\{\\pi(X_1),\\dots, \\pi(X_r)\\}$.\nWe have $\\Aut(F) = \\Aut_{V(F)}(F)$, where $\\Aut(F)$ is the set of all automorphisms of $F$.\n\n\nThe following two lemmas are easy corollaries of the ``rainbow counting lemma'' given in the appendix. Their deduction is also deferred to the appendix.\nRoughly speaking, the proof relies on the fact that the global and local boundedness of the colouring $\\phi$ together imply that the number of non-rainbow copies of  $F$ in $G$ containing a specific vertex or a specific edge is negligible, and so the number of  {\\it rainbow} copies of $F$ in $G$ is roughly the same as the total number of copies of $F$ in $G$.\n\n\n\n\n\n\n\\begin{lemma}\\label{counting partite}\nLet $0<1\/n \\ll \\zeta  \\ll \\eps\\ll d, 1\/r, 1\/C, 1\/f, 1\/h \\leq 1$.\nTake a graph $F$ with $h$ edges and a vertex partition $\\cX=\\{X_1,\\dots, X_r\\}$ of $V(F)$ into independent sets, where $|X_i|=f$.  Take a graph $G$ with a vertex partition $\\cV=\\{V_1,\\dots, V_r\\}$ into independent sets. Suppose that $\\phi$ is a $(C n,\\zeta n)$-bounded colouring of $G$. Fix $j',j''\\in [r]$ and an edge $vw \\in E(G)$ with $v\\in V_{j'}$ and $w \\in V_{j''}$.\nSuppose that the following conditions hold.\n\\begin{enumerate}[label=\\text{{\\rm (A\\arabic*)$_{\\ref{counting partite}}$}}]\n\\item \\label{lem counting partite 2} For each $i\\in [r]$, we have $|V_{i}| = (1\\pm \\zeta )n$.\n\\item \\label{lem counting partite 1} For all $i\\neq j \\in [r]$, the bipartite graph $G[V_{i},V_{j}]$ is $(\\zeta,d)$-superregular.\n\\item \\label{lem counting partite 3} Either\n$d_{G,V_{i}}(v,w) = (d^2 \\pm \\zeta)|V_{i}|$ and $vw \\notin \\Ir_{G}^{\\phi}(\\zeta n)$ for all $i\\in [r]\\setminus\\{j',j''\\}$, or $F$ is triangle-free.\n\\end{enumerate}\nThen for any vertex $u\\in V(G)$, we have\n$$r_{G,\\cX,\\cV}(F,u) = (1\\pm \\eps) \\frac{r! f d^h n^{fr-1}}{|\\Aut_{\\cX}(F)| } \\enspace \\text{ and } \\enspace\nr_{G,\\cX,\\cV}(F,vw) = (1\\pm \\eps) \\frac{  r!h  d^{h-1} n^{fr-2}}{\n\\binom{r}{2} |\\Aut_{\\cX}(F)|}.\n$$\n\\end{lemma}\n\n\n\\begin{lemma}\\label{counting quasirandom}\nLet $0<1\/n \\ll \\zeta  \\ll \\eps\\ll d, 1\/C, 1\/f, 1\/h \\leq 1$.\nTake a graph $F$ with $f$ vertices and $h$ edges and an $n$-vertex graph $G$ which is $(\\zeta,d)$-quasirandom. Suppose that $\\phi$ is a $(C n,\\zeta n)$-bounded colouring of $G$. Fix $vw \\in E(G)$.\nSuppose that the following holds.\n\\begin{enumerate}[label=\\text{{\\rm (A\\arabic*)$_{\\ref{counting quasirandom}}$}}]\n\\item \\label{lem counting quasirandom 1} Either $vw \\notin \\Ir_{G}(\\zeta,d)\\cup \\Ir_{G}^{\\phi}(\\zeta n)$ or $F$ is triangle-free.\n\\end{enumerate}\nThen for any vertex $u\\in V(G)$, we have\n$$r_{G}(F,u) = (1\\pm \\eps\/3) \\frac{f d^h n^{f-1}}{|\\Aut(F)| } \\enspace \\text{ and } \\enspace\nr_{G}(F,vw) = (1\\pm \\eps\/3) \\frac{ 2h d^{h-1} n^{f-2}}{ |\\Aut(F)|}.\n$$\n\\end{lemma}\nNote that in our applications of these lemmas, \\ref{lem counting partite 3} and \\ref{lem counting quasirandom 1} will be satisfied for {\\it all} edges, and thus the conclusion will hold for all edges as well.\n\\subsection{A rainbow blow-up lemma}\n\nThe following statement is an easy consequence of the rainbow blow-up lemma proved by Glock and Joos \\cite{GJ18}, which is our main tool to turn almost-spanning structures into  spanning ones. Note however that the boundedness condition on $\\phi$ is much more restrictive than in our results.\n\n\\begin{theorem}\\label{blowup}\nLet $0<1\/n \\ll \\delta_2 \\ll\\gamma, 1\/r, d, 1\/\\Delta \\leq 1$.\nSuppose that $H$ is a graph with vertex partition $\\{X_0, X_1,\\dots, X_r\\}$ and $G$ is a graph with vertex partition $\\{V_0, V_1,\\dots, V_r\\}$. Let $\\phi$ be a $\\delta_2 n$-bounded colouring of $G$. Suppose that the following conditions hold.\n\\begin{enumerate}[label=\\text{{\\rm (A\\arabic*)$_{\\ref{blowup}}$}}]\n\\item \\label{lem blowup 1} For each $i\\in [r]\\cup \\{0\\}$, $X_i$ is an independent set of $H$ and $\\Delta(H)\\leq \\Delta$. Moreover, no  two vertices of $X_0$ have a common neighbour.\n\\item \\label{lem blowup 2} $\\psi': X_0 \\rightarrow V_0$ is an injective map and $|X_0|\\leq \\delta_2 n$.\n\\item \\label{lem blowup 3} For each $i\\in [r]$, we have $|X_i|\\leq |V_i|$ and $|V_i| = (1\\pm \\delta_2) n$.\n\\item \\label{lem blowup 4} For all $i \\neq j\\in [r]$, the graph $G[V_i,V_j]$ is $(\\delta_2,d)$-superregular.\n\\item \\label{lem blowup 5} For all $x\\in X_0$ and $i\\in [r]$, if\n$N_{H}(x)\\cap X_i \\neq \\emptyset$, then we have $d_{G,V_i}(\\psi'(x)) \\geq \\frac{\\gamma d}2 |V_i|$.\\COMMENT{This is needed for the multi-partite case at the end.}\n\\end{enumerate}\nThen there is an embedding $\\psi$ of $H$ into $G$ which extends $\\psi'$ such that $\\psi(X_i) \\subseteq V_i$ for each $i\\in [r]$  and $\\psi(H)$ is a rainbow subgraph of $G$.\nMoreover, if $|X_i| \\leq (1-\\sqrt{\\delta_2})n$ for all $i\\in [r]$, then the prefix ``super'' in \\ref{lem blowup 4} may be omitted.\n\\end{theorem}\n\n\n\n\\subsection{Matchings in hypergraphs} This section starts with a classical result due to Pippenger and Spencer on matchings in hypergraphs. We then prove a ``defect'' version of this (see Lemma~\\ref{lem: random matching}). We conclude the section with Lemma~\\ref{lem: random F decomp}, which is a translation of  results on almost-spanning matchings in hypergraphs to results on approximate decompositions into rainbow almost-spanning factors. Lemma~\\ref{lem: random F decomp} is an essential step in the proofs of our theorems, allowing to obtain an approximate rainbow structure, which we then complete using the rainbow blow-up lemma.\n\n\n\nRecall that we allow hypergraphs to have multiple edges.  For a hypergraph $\\cH$ and $u,v\\in V(\\cH)$, we let $d_{\\cH}(v):= |\\{H\\in E(\\cH): v\\in H\\}|$ and $d_{\\cH}(uv):= |\\{H\\in E(\\cH): \\{u,v\\}\\subseteq H\\}|$. We let\n$$\\Delta(\\cH):= \\max_{v\\in V(\\cH)} d_{\\cH}(v) \\enspace \\ \\ \\text{and} \\ \\ \\enspace \\Delta_2(\\cH):=\\max_{u\\neq v\\in V(\\cH)} d_{\\cH}(uv) $$\nbe the maximum degree and codegree of $\\cH$, respectively. A {\\it matching} in a hypergraph is a collection of disjoint edges. It is {\\it perfect} if it covers all the vertices of the hypergraph. If all sets in a matching have size $r$, then we call it an {\\it $r$-matching}.\n\n\n\\begin{theorem}\\cite{PS89} \\label{lem: Pippenger}\nLet $0<1\/n\\ll \\epsilon \\ll \\delta, 1\/r <1$.\nIf $\\cH$ is an $n$-vertex $r$-uniform hypergraph satisfying $\\delta(\\cH)\\geq (1-\\eps) \\Delta(\\cH)$ and $\\Delta_2(\\cH) \\leq \\eps \\Delta(\\cH)$, then\n$E(\\cH)$ can be partitioned into $(1+\\delta)\\Delta(\\cH)$ matchings.\n\\end{theorem}\n\nApplying this theorem, we can  prove a variation in which the hypergraph is allowed to have vertices of smaller degree, but the matchings are only required to cover the vertices of ``correct'' degree. We will need the following classical result on resolvable block designs due to Ray-Chaudhuri and Wilson, formulated in terms of matchings of $r$-sets.\n\\begin{theorem}[\\cite{RW73}]\\label{cl: r-factor}\n  For any $r\\in \\mathbb N$ there exists  $b'_0 \\in \\mathbb N,$ such that the following holds for any $b'\\ge b'_0$. For any $\\rho\\le 1$ there exists an $r$-uniform regular hypergraph $\\cA$ on vertex set $X$ of size $b:=r(r-1)b'+r$, such that its degree is $\\lfloor\\rho g\\rfloor$, where $g:=rb'+1=(b-1)\/(r-1)$, and its maximum codegree is $1$. Moreover, $\\cA$ is decomposable into $\\lfloor\\rho g\\rfloor$ perfect $r$-matchings.\n\\end{theorem}\nNote that if we take $\\rho=1$ in the theorem above, then codegree of any two vertices in $X$ is $1$, that is, any pair is contained in exactly one edge of a matching.\nWe now state our ``defect'' version of Theorem~\\ref{lem: Pippenger}.\n\n\n\n\\begin{lemma}\\label{lem: random matching}\nLet $0<1\/n \\ll \\eps\\ll \\delta,1\/r<1$.\nSuppose that $\\cH$ is an $r$-uniform hypergraph satisfying $\\Delta_2(\\cH) \\leq \\eps \\Delta(\\cH)$.\nPut\n$$U:=\\{ u\\in V(\\cH): d_{\\cH}(u) < (1-\\eps )\\Delta(\\cH)\\} \\text{ and }\nV':= V(\\cH)\\setminus U.$$\nSuppose $V\\subseteq V'$ with $|V|=n$.\n\\begin{itemize}\n\\item[(i)]\nThere exist at least $(1-\\delta)\\Delta(\\cH)$ edge-disjoint matchings of $\\cH$ such that each matching covers at least $(1-\\delta)n$ vertices of $V$ and each vertex $v$ of $V$ belongs to at least $(1-\\delta)\\Delta(\\cH)$ of the matchings.\n\n\\item[(ii)]\nThere exists a randomized algorithm which always returns a matching $\\cM$ of $\\cH$ covering at least $(1-\\delta) n$ vertices of $V$  such that for each $v\\in V$ we have\n$$\\mathbb{P}[v\\in V(\\cM)] \\ge 1 -\\delta.$$\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nNote that $\\Delta_2(\\cH)\\geq 1$ implies $\\Delta(\\cH)\\geq \\eps^{-1}$. Before we can apply Theorem~\\ref{lem: Pippenger}, we have to preprocess our hypergraph and make it nearly regular, without increasing the codegree too much. We shall do this in two stages.\n\n\n\nThe first stage is the following process.\nWe iteratively obtain a sequence of hypergraphs\n$\\cH=:\\cH_{0}\\subseteq \\cH_{1}\\subseteq \\dots$ on the same vertex set, until we have that $|U_i|\\le \\eps n$ at some step, where\n \\begin{equation}\\label{eq: smallresidue} U_i:=\\big\\{u\\in V(\\cH): d_{\\cH_i}(u) < (1-\\eps)\\Delta(\\cH)\\big\\}.\\end{equation}\n We additionally require that throughout our process the following hold for each $i$:\n\\begin{align}\\label{eq: cH'}\n\\begin{split}\n\\Delta(\\cH_i) &\\leq (1+\\eps^{1\/3}) \\Delta(\\cH), \\enspace \\Delta_2(\\cH_i) \\leq \\Delta_2(\\cH)+ 2i\\ \\ \\ \\  \\text{ and } \\\\\n\\delta(\\cH_i) &\\geq \\min\\{ (1-\\eps)\\Delta(\\cH),\n\\delta(\\cH) + \\eps^{-1\/2} i\\}.\n\\end{split}\n\\end{align}\nNote that $\\cH_0$ satisfies \\eqref{eq: cH'}. Suppose that we have constructed $\\cH_{i}$ and assume that $|U_i|> \\eps n$. Then we find two (not necessarily disjoint) sets $U^1_i$ and $U^2_i$ of the same size $r(r-1)b'+r$ for some integer $b'$, such that $U^1_i\\cup U^2_i = U_i$. We apply Theorem~\\ref{cl: r-factor} to $U_i^j$, $j\\in [2]$, and find an $r$-uniform regular hypergraph $\\cA_i^j$ on $U_i^j$ with degree  $\\eps^{-1\/2}$ and maximum codegree $1$. Then we put $\\cH_{i+1}:=\\cH_i\\cup \\cA_i^1\\cup\\cA_i^2$, and repeat the procedure until   $|U_i|\\le \\eps n$ for some $i$, say $i=k$. Note that we may well be adding the same edge multiple times and, should this be the case, keep multiple copies of it.\n\nLet us verify the validity of \\eqref{eq: cH'}. Clearly, the minimum degree increases by at least $\\eps^{-1\/2}$ at each step, while the codegree of any two vertices increases by at most $2$. We also have\n$$\\Delta(\\cH_{i+1}) \\leq \\max\\{\\Delta(\\cH_{i}), (1-\\eps)\\Delta(\\cH) + 2 \\eps^{-1\/2} \\} \\leq  (1+\\eps^{1\/3}) \\Delta(\\cH).$$\n\nRecall that $|U_k|\\le \\eps n$ and $k$ is the smallest index for which it holds. Due to the minimum degree condition in \\eqref{eq: cH'}, we have $k\\le \\eps^{1\/2}\\Delta(\\cH)$, and thus\n\\begin{eqnarray}\\label{eq: cH'2}\n\\Delta_2(\\cH_k)\\stackrel{\\eqref{eq: cH'}}{\\leq} 3 \\eps^{1\/2}\\Delta(\\cH).\n\\end{eqnarray}\nThis concludes the first stage of modification.\n\n\nThe goal of the second stage is to fix the degrees in the small exceptional set $U_k$. Put $t:= \\sum_{u\\in U_k} (\\Delta(\\cH) - d_{\\cH_k}(u))$. Note that $t\\le \\eps n\\Delta(\\cH)$.  Consider a family $\\cW$ of disjoint $(r-1)$-sets on $V(\\cH)\\setminus U_k$, such that\n$$|\\cW|= \\Big\\lfloor\\frac{|V(\\cH)\\setminus U_k|}{r-1}\\Big\\rfloor > \\frac nr.$$\nConsider an $r$-uniform hypergraph $\\mathcal B$ on $V(\\cH)$, such that each edge of $\\mathcal B$ has the form $\\{u\\}\\cup W$, where $u\\in U_k$, $W\\in \\cW$, and, moreover, each $u$ is contained in exactly $\\Delta(\\cH) - d_{\\cH_k}(u)$ edges of $\\mathcal B$ and each $W$ is contained in at most\n$\\lceil\\frac {t}{|\\cW|}\\rceil \\le r\\eps\\Delta(\\cH)$ edges. Note that $\\Delta_2(\\cB)\\le r\\eps \\Delta(\\cH)$, as well as that for any $v\\in V(\\cH)\\setminus U_k$ we have $d_{\\cB}(v)\\le r\\eps \\Delta(\\cH)$. Consider the hypergraph $\\cG:=\\cH_k\\cup \\cB$. Then, clearly, $\\delta(\\cG)\\ge (1-\\eps)\\Delta(\\cH)$, but also\n\\begin{alignat*}{2}\\Delta(\\cG) &\\leq \\Delta(\\cH_k) + r\\eps \\Delta(\\cH)\n&&\\stackrel{\\eqref{eq: cH'}}{\\leq} (1+\\eps^{1\/4})\\Delta(\\cH),\\\\\n \\Delta_2(\\cG)&\\leq \\Delta_2(\\cH_k)+ r\\eps \\Delta(\\cH)\n &&\\stackrel{\\eqref{eq: cH'2}}{\\leq }  \\eps^{1\/4}\\Delta(\\cH).\\end{alignat*}\n\nSince $\\eps \\ll \\delta$, we are now in a position to apply Theorem~\\ref{lem: Pippenger}, and decompose $\\cG$ into a family $\\mathcal F''$ of $(1+\\delta^{5})\\Delta(\\cG)$ matchings. At least $(1-\\delta^2)\\Delta(\\cH)$ of these matchings must cover at least $(1-\\delta^2)n$ vertices of $V$.\n\nLet $\\mathcal F'$ denote the family of these almost-spanning matchings and let $\\cF:=\\{\\cM\\cap E(\\cH):\\cM\\in \\cF'\\}$. We claim that the collection $\\cF$ satisfies the assertion of the first part of the lemma, moreover, the algorithm which chooses one of the matchings from $\\mathcal F$ uniformly at random and returns its intersection with $\\mathcal H$ satisfies the assertion of the second part of the lemma.\n\nTo see this, first note that any matching $\\cM\\in \\mathcal F'$ covers at least $(1-\\delta^2)n$ vertices of $V$, and, since any edge in $\\cG-\\cH$\neither entirely lies in $U$ or intersects $U_k$, the matching $\\cM\\cap E(\\cH)$ covers at least $(1-\\delta^2)n - r|U_k| \\ge (1-\\delta) n$ vertices of $V$.\nSecond, for each  $v\\in V(\\cH)$, the vertex $v$ belongs to $d_{\\cH}(v) \\pm |\\mathcal{F''}\\setminus \\mathcal{F'}|\\ge (1-\\delta)\\Delta(\\cH)$ matchings from $\\mathcal{F'}$ that cover $v$ by an edge from $\\cH$. This proves (i).\nTo prove (ii), note that $|\\mathcal F| = |\\cF'|=(1\\pm \\delta^2)\\Delta(\\cH)$, hence for a randomly chosen $\\cM\\in \\mathcal F$, for each $v\\in V$, we have\n$$\\mathbb{P}[v\\in V(\\cM) ] = \\frac{ d_{\\cH}(v) \\pm |\\mathcal{F''}\\setminus \\mathcal F'| }{ |\\mathcal{F}| } = \\frac{d_{\\cH}(v)}{|\\mathcal F| } \\pm 3\\delta^2\n\\geq  1- \\delta.$$\n\\end{proof}\n\n\n\nFor an edge-coloured graph $G$ and a given family $\\cF$ of rainbow subgraphs of $G$, we denote by $\\cF(v_1,v_2;c_1,c_2)$ the subfamily of all those graphs from $\\cF$ which contain the vertices $v_1,v_2$ and edges of colours $c_1,c_2$. We define $\\cF(v_1;c_1)$, $\\cF(v_1,v_2)$, $\\cF(c_1,c_2)$, $\\cF(v_1)$ and $\\cF(c_1)$ in a similar way. For $uw, u'w'\\in E(G)$ we denote by $\\cF(uw)$ the subfamily of graphs from $\\cF$ that contain the edge $uw$, and define $\\cF(uw,u'w')$ similarly. The next lemma is the key to the proof of Theorem~\\ref{thm: approx decomp} and is also very important for the proofs of the other theorems from Section~\\ref{sec2}.\n\n\\begin{lemma}\\label{lem: random F decomp}Let $0<1\/n \\ll \\eps\\ll \\delta, 1\/f \\le 1$.\nSuppose that $F$ is a graph on $f\\ge 3$ vertices and with $h\\ge 1$ edges. Suppose that $G=(V,E)$ is an $n$-vertex graph and $\\phi$ is an $m$-colouring of $G$. Consider a family $\\cF$ of rainbow copies of $F$ in $G$ that satisfies the following requirements.\n \\begin{enumerate}[label=\\text{{\\rm (A\\arabic*)$_{\\ref{lem: random F decomp}}$}}]\n \\item\\label{F decomp 1} For any $v,v_1,v_2\\in V$ and $c_1,c_2\\in [m]$ we have $$\\max\\{|\\cF(v_1,v_2)|,|\\cF(v_1;c_1)|, |\\cF(c_1,c_2)|\\} \\leq \\eps|\\cF(v)|.$$\n \\item\\label{F decomp 2} For any $c\\in [m]$ and $v\\in V$ we have $|\\cF(v)|\\ge (1-\\eps)|\\cF(c)|$.\n\\item \\label{F decomp 3}\nFor all $v\\in V$ and $uw\\in E$ we have \\COMMENT{Note that $\\cF(uw)\\subset \\cF(u,w)$ and thus $t$ is large due to \\ref{F decomp 1} provided $e(G)>0$.}\n\\begin{align*}\n(1\\pm \\eps)\\frac{|\\cF(v)|}{|\\cF(uw)|}=\\frac{f|E|}{h|V|}=:t.\n\\end{align*}\n\\item\\label{F decomp 4} For any $uw\\in E$ we have $|\\cF(uw)|\\ge 10\\eps^{-1}\\log n$.\n\\item\\label{F decomp 5} For any $uw,u'w',u''w''\\in E$ we have $\\eps|\\cF(uw)|\\ge |\\cF(u'w',u''w'')|$.\n\n\\end{enumerate}\nThen there exists a randomized algorithm which always returns  $(1-\\delta)t$ edge-disjoint  rainbow $\\delta$-spanning $F$-factors $\\cM_1,\\ldots, \\cM_{(1-\\delta)t}$ of $G$, such that each $\\cM_i$ consists of copies of $F$ from $\\cF$ and for all $v\\in V$ and $i\\in[(1-\\delta)t]$ we have\n$$\\mathbb{P}[ v\\in V(\\cM_i)] \\ge 1 -\\delta.$$\n\\end{lemma}\nClearly, the union of all the $\\cM_i$ covers all but at most a $2\\delta$-proportion of edges of $G$.\\COMMENT{The total number of edges covered is $(1-\\delta)\\frac{f|E|}{h|V|}\\cdot \\frac{(1-\\delta)h|V|}{f}=(1-\\delta)^2|E|.$}\n\n\\begin{proof}\nThe idea is to apply Lemma~\\ref{lem: random matching} (ii) to a suitable auxiliary (multi-) hypergraph $\\cH$. However, the choice of $\\cH$ is not straightforward, since Lemma~\\ref{lem: random matching} (ii) gives only a single random matching while we need an almost-decomposition. We can resolve this by turning both the edges and the vertices of $G$ into vertices of $\\cH$. However, this gives rise to the issue that the potential degrees of vertices and edges in the corresponding auxiliary hypergraph are very different. This in turn can be overcome by the following random splitting process.\n\nConsider a random partition $\\cF_1,\\dots, \\cF_t$ of $\\cF$ into $t$ parts, where for all $\\bar F\\in \\cF$ and $i\\in [t]$  we have $\\bar F \\in \\cF_i$ with probability $1\/t$ independently of all other graphs in $\\cF$.  Using Lemma~\\ref{Chernoff} combined with the fact that the expected value of $\\cF(v)\\cap \\cF_i$ is sufficiently large (it is at least $9\\eps^{-1}\\log n$ by \\ref{F decomp 4} and \\ref{F decomp 3}) for each $v\\in V(G)$, we can conclude that for any $uw\\in E,\\ v_1,v_2\\in V$, $c,c_1,c_2\\in [m]$ and $i,i'\\in[t]$ we have\n\\COMMENT{Note that $\\EXP[|\\cF(v)\\cap\\cF_i|]=\\frac 1t|\\cF(v)|\\ge 9\\eps^{-1}\\log n$. Using Lemma~\\ref{Chernoff}, $$\\Pro\\big[\\big||\\cF(v)\\cap\\cF_i|-\\EXP[|\\cF(v)\\cap\\cF_i|]\\big|\\ge 2\\eps^{1\/2}\\EXP[|\\cF(v)\\cap\\cF_i|]\\big]\\le e^{-\\eps \\EXP[|\\cF(v)\\cap\\cF_i|]}<n^{-9}.$$\nSince $\\EXP[|\\cF(c)\\cap \\cF_i|]\\le (1+\\eps) \\EXP[|\\cF(v)\\cap \\cF_i|]$ by \\ref{F decomp 2}, we also have\n$$\\Pro\\big[|\\cF(c)\\cap\\cF_i|-\\EXP[|\\cF(c)\\cap\\cF_i|]\\ge 2\\eps^{1\/2}\\EXP[|\\cF(v)\\cap\\cF_i|]\\big]\\le e^{-\\eps \\EXP[|\\cF(v)\\cap\\cF_i|]}<n^{-9}.$$\nTogether these two formulas imply that all \\eqref{cond decomp 1} hold simultaneously with high probability.\n\nSimilarly, any of the families $\\cF(v_1,v_2)\\cap \\cF_{i'},\\ \\cF(v_1;c_1)\\cap \\cF_{i'},\\ \\cF(c_1,c_2)\\cap \\cF_{i'}$ in expectation have size at most $\\eps \\EXP[|\\cF(v)\\cap\\cF_i|]$. Thus, the probability that they surpass their expectation by $2\\eps^{1\/2} \\EXP[|\\cF(v)\\cap\\cF_i|]$ is at most $e^{-\\eps^{1\/2}\\EXP[|\\cF(v)\\cap\\cF_i|]}<n^{-9}.$ Thus, all inequalities \\eqref{cond decomp 2} hold simultaneously with high probability.\n}\n\\begin{align}\n   \\label{cond decomp 1}(1\\pm 3\\eps^{1\/2}) |\\cF(uw)|\\overset{\\ref{F decomp 3}}{=}&|\\cF(v)\\cap\\cF_i |\\overset{\\ref{F decomp 2}}{\\ge} (1-5\\eps^{1\/2})|\\cF(c)\\cap \\cF_{i'}|, \\\\\n   \\label{cond decomp 2} 3\\eps^{1\/2} |\\cF(v)\\cap\\cF_i |\\overset{\\ref{F decomp 1}}{\\ge}& \\max\\{|\\cF(v_1,v_2)\\cap \\cF_{i'}|,|\\cF(v_1;c_1)\\cap \\cF_{i'}|, |\\cF(c_1,c_2)\\cap \\cF_{i'}|\\}.\n\\end{align}\n\nConsider the hypergraph $\\cH$ defined by\n\\begin{align}\nV(\\cH)&:=E\\cup (V\\times [t]) \\cup ([m]\\times[t])\\ \\ \\ \\ \\ \\ \\ \\ \\ \\  \\text{and} \\enspace \\notag\\\\\n\\label{colorhgraph} E(\\cH)&:= \\big\\{E(\\bar F)\\cup (V(\\bar F)\\times \\{i\\})\\cup (\\phi(E(\\bar F))\\times \\{i\\}) : \\bar F\\in \\cF_i, i\\in [t] \\big\\}.\n\\end{align}\n\nThus each edge of $\\cH$ corresponds to some $\\bar F\\in\\cF$. Condition \\eqref{cond decomp 1} guarantees that the vertices from $E\\cup (V\\times [t])$ have roughly the same degree,\\COMMENT{\nThis  is clear from \\eqref{cond decomp 1} itself, since we can substitute different values of $v, u,w,c$.\n}\nwhile the vertices from $[m]\\times [t]$ have degree that is at most $(1-8\\eps^{1\/2})^{-1}$ times the degrees of any vertex from $E\\cup (V\\times [t])$, but may be significantly smaller. Moreover, we have $\\Delta_2(\\cH)\\le 3\\eps^{1\/2} \\Delta(\\cH)$ due to \\eqref{cond decomp 2} and \\ref{F decomp 5} (note here that vertices of $\\cH$ that have different indices $i,i'\\in[t]$ have zero codegree and that $|\\cF(uw)|\\le |\\cF(u,w)|$ for any $uw\\in E$). Therefore, we can apply Lemma~\\ref{lem: random matching}~(ii) with $8\\eps^{1\/2}, \\delta^3$ and $V\\times[t]$ playing the roles of $\\eps, \\delta$ and $V$, respectively and obtain an algorithm producing a random matching $\\cM$ of $\\cH$, covering at least a $(1-\\delta^3)$-proportion of vertices from $V\\times [t]$ and such that each vertex of $V\\times [t]$ is contained in $\\cM$ with probability at least $1-\\delta^3$. In particular, this implies that\n\\begin{equation}\\label{eqspan t} \\cM\\text{ covers all but at most a }\\delta\\text{-proportion of }V\\times \\{i\\}\\text{ for at least }(1-\\delta)t\\text{ values of }i.\n\\end{equation}\n\nLet $\\cM_i'$ be the collection of all those $\\bar F\\in \\cF_i$ which correspond to some edge of $\\cM$. Then $\\cM_i'$ forms a rainbow $c_i$-spanning $F$-factor in $G$ for some $c_i>0$, moreover, these factors are edge-disjoint for different $i_1,i_2\\in[t]$. Furthermore, by Lemma~\\ref{lem: random matching} for each $v\\in V$ we have $\\Pro[v\\in V(\\cM_i')]\\ge 1-\\delta^3$. However, $\\cM'_i$ does not necessarily form a $\\delta$-spanning $F$-factor.\n This can be fixed easily.  Since for each $i\\in [t]$ the matching $\\cM$ covers each vertex from $V\\times \\{i\\}$ with probability $1-\\delta^3$, for each $i$ with probability at least $1-\\delta\/2$ the factor $\\cM_i'$ is $\\delta$-spanning,\\COMMENT{\n Suppose not, i.e. there exists a set of $>\\delta$-spanning (``bad'') matchings with probability to appear $\\ge \\delta\/2$. $$\\Pro[x \\text{ is not covered}]\\ge \\Pro[x \\text{ is not covered}\\mid \\cM_i\\text{ is bad}]\\Pro[\\cM_i\\text{ is bad}]\\ge \\frac{\\delta}2\\Pro[x \\text{ is not covered}\\mid \\cM_i\\text{ is bad}].$$\n Averaging over the choice of $x$, the definition of ``bad'' matchings implies that $\\mathbb E[\\text{proportion of }x\\in V\\text{ which are not covered}\\mid \\cM_i\\text{ is bad}]\\ge \\delta$. Thus, on average $\\Pro[x \\text{ is not covered}]\\ge\\delta^2\/2$, which is a contradiction.\n }\n and thus with probability at least $1-\\delta$ it is both $\\delta$-spanning and covers a given vertex $v$.\\COMMENT{\nNote that we cannot simply return ``the first $(1-\\delta)t$  $\\cM_i$ that are $\\delta$-spanning'', since we would not be able to guarantee that, once we fix a choice of $j$, the $j$-th matching in this decomposition would cover each vertex with probability at least $1-\\delta$. The problem is that the $j$-th matching doesn't necessarily contain all large matchings $\\cM_j$, and is overall a composite of matchings from different layers (values of $i\\in[t]$). Thus, while having a large size every time, we loose the control over the probability that a given vertex is contained in the matching with high probability.\n}\nMoreover, the factors  $\\cM_i'$ are $\\delta$-spanning for at least $(1-\\delta)t$ values of $i$ (cf. \\eqref{eqspan t}).  Let the algorithm return the factors $\\cM_i$, $i\\le (1-\\delta)t$, where $\\cM_i:=\\cM_i'$ if $\\cM_i'$ is $\\delta$-spanning, and otherwise $\\cM_i:=\\cM_j'$ for some $j> (1-\\delta)t$, where $\\cM_j'$ is a $\\delta$-spanning factor not yet used to substitute for $\\cM_{i'}$ with $i'<i.$ Note that for any given $v\\in V$ and any $i\\in[(1-\\delta)t]$, we have $\\Pro[v\\in V(\\cM_i)]\\ge \\Pro[v\\in V(\\cM_i)\\wedge \\cM_i = \\cM_i']\\ge 1-\\delta$ as required.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{Partitions} To have better control over the colours and vertices used when constructing the decompositions, we need to split vertices and colours into groups. The results in this section will be needed to ensure that the relevant properties of the original graph are inherited by its subgraphs induced by suitable random partitions.\n\n\nWe say that $\\cV$ is \\emph{a partition of a set $V$ chosen at random with  probability distribution $(p_1,\\dots, p_r)$}, if $p_1,\\dots, p_r$ are nonnegative real numbers satisfying $\\sum_{i\\in [r]} p_i\\le 1$ and $\\cV$ is a random variable such that for each $v\\in V$, we have $\\mathbb{P}[v\\in V_i] = p_i$ independently at random.\n\n\n\\begin{lemma}\\label{qr to sr}\nLet $0<1\/n \\ll \\eta \\ll\\zeta \\ll \\delta, d, 1\/C\\leq 1$.\nLet $G'$ be a $(\\eta,d)$-quasirandom $n$-vertex graph. Suppose that $\\phi$ is a $(C n,\\eta n)$-bounded $m$-colouring of $G'$. Then there exists a $(\\zeta,d)$-quasirandom spanning subgraph $G$ of $G'$, such that for any $uv\\in E(G)$ we have\n\\begin{equation}\\label{qr 0} d_{G}(u,v)= (d^2\\pm \\zeta) n \\enspace \\text{and} \\enspace c^{\\phi}_{G}(u,v)  \\leq \\zeta n.\\end{equation}\nMoreover, the following holds.\nFor a random partition  $\\cV$ of $V(G)$ with probability distribution $(p_1,\\dots, p_r)$, where $p_i\\ge n^{-1\/2}$ for each $i\\in[r]$, \\COMMENT{The sentence ``and $p_i\/p_j\\ge\\delta$ for each $i,j\\in [r]$,'' was here. }\nwith probability at least $0.9$ we have:\n \\begin{enumerate}[label=\\text{{\\rm (A\\arabic*)$_{\\ref{qr to sr}}$}}]\n\\item \\label{qr 1} $|V_i|= (1\\pm \\zeta)p_i n$.\n\\item \\label{qr 2} For all $i\\neq j \\in [r]$, the bipartite graph $G[V_{i},V_{j}]$ is $(\\zeta,d)$-superregular.\n\\item \\label{qr 3} For all $vw\\in E(G)$ and $i\\in [r]$, we have $|C_G^{\\phi}(v,w)\\cap V_i|\\le \\zeta |V_i|$ and\n$d_{G,V_{i}}(v,w) = (d^2 \\pm \\zeta)|V_{i}|$.\n\\item \\label{qr 4} For all $i\\in [r]$, the graph $G[V_i]$ is $(\\zeta,d)$-quasirandom.\n\\end{enumerate}\n\\end{lemma}\nNote that $G$ contains all but at most a $2\\zeta\/d$-fraction of the edges of $G'$.\n\\begin{proof} Let $\\Ir$ be the graph with vertex set $V(G')$ and edge set $E(\\Ir_{G'}(\\eta^{1\/10},d))\\cup E(\\Ir^{\\phi}_{G'}(\\eta^{1\/2} n))$, that is, every edge of $\\Ir$ corresponds to a pair of vertices which either has ``wrong'' codegree or ``wrong'' monochromatic codegree. We first show that $\\Ir$ has small maximum degree.  Since $G'$ is a $(\\eta,d)$-quasirandom graph and $\\phi$ is locally $\\eta n$-bounded, by Lemmas~\\ref{lem: irregular degree} and~\\ref{lem: colour irregular degree} we have\n\\begin{align}\\label{eq: irregularity graphs}\n\\Delta(\\Ir) \\leq \\Delta(\\Ir_{G'}(\\eta^{1\/10},d)) + \\Delta(\\Ir_{G'}^{\\phi}(\\eta^{1\/2} n))\n\\leq \\eta^{1\/10} n + \\eta^{1\/2} n \\le \\zeta^{10} n.\n\\end{align}\nConsider the graph $G:=G'-E(\\Ir)$. For each $uv\\notin \\Ir$ (and so in particular for each $uv \\in E(G)$), we have\n\\begin{align}\\label{eq: dc value irr}\nd_{G}(u,v)= (d^2\\pm \\zeta^{2}) n \\enspace \\text{and} \\enspace c^{\\phi}_{G}(u,v)  \\leq \\zeta^{2} n.\n\\end{align}\n Clearly, $G$ is $(\\zeta^2,d)$-quasirandom, and \\eqref{eq: dc value irr} implies \\eqref{qr 0}.  Now, a standard application of Chernoff bounds (Lemma~\\ref{Chernoff}) implies that \\ref{qr 1} and \\ref{qr 3} hold with probability $0.99$. For the same reasons, $d_{G,V_{i}}(v) = (1\\pm \\zeta\/2)p_i d_{G}(v)$ for all $v\\in V(G), i\\in [r]$ with probability $0.99$ (this implies the ``super'' part of the superregularity from \\ref{qr 2}).\\COMMENT{So far, we were simply partitioning polynomially many sets (``neighbors of $v$ in $V_i$'', ``common neighbors of $v,w$ connected by edges of the same colour'' etc) and claiming that all of them have high probability to have size close to the expectation. Just note that in expectation they are (roughly) at least $p_in\\ge n^{1\/2}$, so the calculations will work easily. It is essentially the same for the two other applications.} Note that $p_i\\ge n^{-1\/2}$ implies that $r\\le n^{1\/2}$.\n\n Finally,  by \\eqref{eq: irregularity graphs} and Lemma~\\ref{Chernoff}, with probability $0.99$ for all $i,j\\in [r]$ the maximum degree of  a vertex from $V_i$ in $\\Ir[V_i,V_j]$ is at most $\\zeta^7p_jn$,\\COMMENT{For a change, I explain this application. All the others are done similarly. The expected degree of any vertex from $V_i$ in this graph is at most $\\zeta^{10}|V_j|$. Since all the vertices are included in $V_j$ independently and using Lemma~\\ref{Chernoff}, the chances that it depasses 7 times the expected value is at most $e^{-\\zeta^{10}|V_j|}<n^{-10}$.} and thus the number of edges of $\\Ir$ that connect vertices of $V_i$ is at most $\\zeta^7 p_i n |V_i| \\leq \\zeta^6 |V_i|^2$ for all $i\\in [r]$.\\COMMENT{Old text: ``the number of edges of $\\Ir$ that connect vertices of $V_i$ and $V_j$ is at most  $\\zeta^7p_jn|V_i|\\le \\zeta^6\\min\\big\\{|V_i|^2,|V_j|^2\\big\\}$.''\n Now it's changed into the current one. Note that, for applying Theorem~\\ref{thm: almost quasirandom}, we need to consider the irregular pairs of vertices which entirely lies in either $V_i$ or $V_j$. We don't care the codegree of $u$ and $v$ where $u\\in V_i$ and $v\\in V_j$.\n Because of this change, the sentence `$p_i\/p_j \\geq \\delta$' at the statement is gone.\n } Applying Theorem~\\ref{thm: almost quasirandom}, we obtain that $G[V_i,V_j]$ is $(\\zeta,d)$-regular. One can similarly bound the number of irregular pairs in each $G[V_i]$, and, combined with the bounds for the degrees and codegrees obtained above, it follows that \\ref{qr 4} holds with  probability at least $0.99$. Overall, all these events hold simultaneously with probability at least $0.9$.\n\\end{proof}\n\n\n\n\\begin{lemma} \\label{lem: graph partition}\nLet $n,r\\in \\mathbb{N}$ and $0<1\/n \\ll \\zeta\\ll 1\/C <1$. Assume that $\\ell\\ge \\frac{n}{\\log^3n}$.\nSuppose that $G$ is an $n$-vertex graph with at most $Cn$ edges and $\\Delta(G)\\leq \\ell$, and $\\cV=(V_1,\\dots, V_r)$  is a partition of $V(G)$ chosen at random with probability distribution $(p_1,\\dots, p_r)$ with $p_i\\geq \\log^{-2}{n}$. Let $p:= \\min_{i\\in [r]}\\{ p_i\\}$.\n  Then with probability at least $1-2r^2e^{-\\frac{\\zeta^3p n}{\\ell}}$ we have for all $i\\ne j\\in [r]$\n $$ e(G[V_i,V_{j}]) =  2p_ip_j  e(G)\\pm \\zeta p_ip_j n \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ e(G[V_i]) = p_i^2e(G)\\pm \\zeta p_i^2 n.$$\n\\end{lemma}\n\\begin{proof}\nLet $v_1,\\ldots, v_n$ be the vertices of $G$ in the decreasing degree order. Put $t:= 2Cn^{1\/3}$. Since $e(G)\\le Cn$, we have   $d(v_k)\\le n^{2\/3}$ for $k> t$. Fix $i,j\\in[r]$. We now count edges with the first (in the ordering) vertex in $V_i$ and the second in $V_j$. We denote this quantity by $\\vec{e}(G[V_i,V_j])$.   Consider a martingale $X_0,\\ldots, X_n$, where $$X_k:=\\Exp\\big[\\vec e(G[V_i,V_j])\\mid V_i\\cap \\{v_1,\\ldots, v_k\\}, V_j\\cap \\{v_1,\\ldots, v_k\\}\\big].$$\nWe aim to apply Theorem~\\ref{Azuma2} to this martingale.  In the notation of that theorem, for $k>t$, we clearly have $|X_k-X_{k-1}|\\le d(v_k)\\le n^{2\/3}$. Moreover, $\\sum_{k=t}^n\\sigma^2_k\\le \\sum_{k=t}^nd^2(v_k)\\le n^{2\/3}\\sum_{k=t}^n d(v_k)\\le 2Cn^{5\/3}$.\n\n\n We now suppose that $k\\le t$. Without loss of generality, we assume that there are no edges in $G$ between vertices $v_{k}, v_{k'}$ for $k,k'\\le t$. Indeed, this accounts for at most $N:=4C^2n^{2\/3}$ edges, which is negligible, and we will take care of this later.\n Take  $k\\le t$ and fix $V_i\\cap \\{v_1,\\ldots, v_{k-1}\\}, V_j\\cap \\{v_1,\\ldots, v_{k-1}\\}$. Then $X_{k}-X_{k-1}$ is the following  random variable:\\COMMENT{For fixed $V_i\\cap \\{v_1,\\ldots, v_{k-1}\\}, V_j\\cap \\{v_1,\\ldots, v_{k-1}\\}$, $X_{k-1}$ is simply a constant equal to $d_1 p_j+d_2 p_ip_j+d(v_k)p_ip_j$, where $d_1$ is the sum of the degrees of vertices from $V_i\\cap \\{v_1,\\ldots, v_{k-1}\\}$ (recall that no edges go between the first $t$ vertices),  $d_2$ is the total number of edges in $G-\\{v_1,\\ldots, v_{k}\\}$. Similarly, $X_k$ has the first two terms, but the third term is replaced by an indicator random variable: $X_k =d_1 p_j+d_2 p_ip_j+\\chi_kd(v_k)p_j$, where $\\chi_k=1$ iff $v_k\\in V_i$. From here we obtain the displayed formula for $X_k-X_{k-1}$.}\n$$X_{k}-X_{k-1}=\\begin{cases}\n                              p_j(1-p_i)d(v_k), & \\mbox{if } v_{k}\\in V_i \\\\\n                                         -p_ip_jd(v_k) & \\mbox{otherwise}.\n                                       \\end{cases}$$\nFrom this formula we can easily conclude that, first, $|X_k-X_{k-1}|\\leq p_jd(v_k)\\le p_j \\ell$, and, second, $\\mathrm{Var}[X_k\\mid X_{k-1},\\ldots, X_1]= \\EXP[(X_k-X_{k-1})^2\\mid X_{k-1},\\ldots, X_1]\\le p_ip_j^2d^2(v_k)=:\\sigma_k^2$.\\COMMENT{\n$\\EXP[(X_k-X_{k-1})^2]\\le p_i(p_j(1-p_i)d(v_k))^2 +(1-p_i)(p_ip_jd(v_k))^2\\le p_ip_j^2d^2(v_k)$\n}\nThus, $\\sum_{k=1}^{t}\\sigma^2_k \\le 2C p_ip_j^2n\\ell.$ Altogether, with $\\vartheta$ defined as in Theorem~\\ref{Azuma2}, we have\n$$\\sum_{k=1}^{n}\\sigma^2_k \\le 2Cp_ip_j^2n\\ell+2Cn^{5\/3}\\le 3Cp_ip_j^2\\ell n \\ \\ \\ \\text{and }\\ \\ \\ \\ \\ \\vartheta\\le \\max\\{p_j \\ell,n^{2\/3}\\}\\le p_j\\ell.$$\n(This is the only place where we make use of the lower bound on $\\ell$.) Substituting into Theorem~\\ref{Azuma2}, we obtain \\begin{align}\\label{eq216}\\mathbb P\\Big[|X_k-p_ip_je(G)|\\le \\frac{\\zeta}3 p_ip_j n\\Big]\\le & 2\\exp\\Big(-\\frac{\\big(\\frac{\\zeta}3 p_ip_j n\\big)^2}{6Cp_ip_j^2\\ell n+\\frac{\\zeta}3 p_ip_jn\\cdot p_j\\ell}\\Big)\\le 2e^{-\\frac{\\zeta^3p_in}{\\ell}}.\\end{align}\nNote that $N\\le \\frac{\\zeta}{6} p_ip_j n$, and thus \\eqref{eq216}, with $\\zeta\/3$ replaced by $\\zeta\/2$ on the left hand side, also holds in the situation when we may have edges between $v_k,\\ v_{k'}$ for $k,k'\\le t$. The fact that $e(G[V_i]) = \\vec e(G[V_i,V_i])$ and $e(G[V_i,V_j]) = \\vec e(G[V_i,V_j])+ \\vec e(G[V_j,V_i])$ if $i\\ne j$, together with a union bound over all possible choices of $i,j\\in [r]$, implies the result.\\end{proof}\n\nThe next lemma allows us to extend the counting results of Lemmas~\\ref{counting partite} and~\\ref{counting quasirandom} to the case when the graph is sparse.\n\\begin{lemma}\\label{lem: colour partition}\nLet $n,r\\in \\mathbb{N}$ and $0<1\/n \\ll \\eps\\ll 1\/f, 1\/C <1$. Assume that $\\ell\\ge n^{2\/3}$.\nSuppose that $G$ is an $n$-vertex graph and $\\phi$ is a $(Cn,\\ell)$-bounded $m$-colouring of $G$. Fix a $k$-vertex subset $U$ of $V(G)$.\nSuppose that $\\cI=(I_1,\\dots, I_r)$ is a partition of $[m]$ chosen at random with probability distribution $(p_1,\\dots, p_r)$, where $p_i\\ge \\log^{-2}n$.\nSuppose that $\\cF$ is a collection of $f$-vertex $h$-edge rainbow subgraphs of $G$ such that $U$ is an independent set of each $R\\in \\cF$. Assume that, for some $a\\ge 1$, the set $U$ has at most $a$ edges incident to it in each $R\\in\\cF$. For $j\\in [r]$, with probability at least  $1- 2\\exp\\Big(-\\frac{\\eps^{4} p_j^{2a-1} n}{\\ell}\\Big)$ the number of graphs $R$ in $\\cF$\nwhich are subgraphs of $G(\\phi,I_j)$ is\n$ p_j^{h} |\\cF|\\pm \\eps p_j^hn^{f-k}.$\n\\end{lemma}\n\\begin{proof} The proof of this lemma is similar to that of Lemma~\\ref{lem: graph partition}.\nFix $j\\in [r]$ and let $L$ be the random variable equal to the number of graphs $R \\in \\cF$ such that the colour of every edge of $R$ belongs to $I_j$. As $R$ contains $h$ edges whose colours are all different, we have $\\mathbb{E}[L] = p_j^{h} |\\cF|$. Order the colours in $[m]$ by the number of graphs $R\\in \\cF$ that contain that colour, from the larger value to the smaller.  Put $t:=hf!n^{1\/2}$. The number of $R\\in \\cF$ that contain some edge $e$ of colour $i\\le t$, where $e$ is not adjacent to one of the vertices of $U$, is at most $t\\cdot Cn \\cdot f!n^{f-k-2}\\le n^{f-k-1\/3}$. We assume for the moment that there are no such $R$, and will deal with them later.\n\n\nFor each $i\\in [m]$, we let $X_i = \\mathbb{E}[L \\mid I_j \\cap [i]]$. Then $X_0,X_1,\\dots, X_m$ is an exposure martingale. Let $C_{i}$ be the number of $R\\in\\cF$ which contain an edge of colour $i$. Let $C_i(j)$ be  the number of $R\\in \\cF$ that are coloured with colours from $I_j$ and which contain an edge of color $i$. It is easy to see that\\COMMENT{If $C_t> n^{f-k-1\/2}$, then $\\sum_{i=1}^tC_i\\ge t n^{f-k-1\/2}>hf!n^{f-k}\\ge h|\\cF|$, which is impossible.}  for $i\\ge t$ we have $C_i\\le n^{f-k-1\/2}$. This implies that $|X_{i}-X_{i-1}|\\le n^{f-k-1\/2}$ for $i\\ge t$, moreover, in the notation of Theorem~\\ref{Azuma2}, $\\sum_{i=t}^{m}\\sigma_i^2\\le \\sum_{i=t}^m C_i^2\\le C_t\\cdot h|\\cF|\\le h f! n^{2f-2k-1\/2}$.\n\nTake $i<t$ and fix $I_j \\cap [i-1].$ Then the random variable $X_i-X_{i-1}$ has the following form:\\COMMENT{By analogy with the previous lemma, the difference of $X_i$ and $X_{i-1}$ are only in graphs from $\\cF$ that contain colour $i$. Note also the definition of $C_i(j)$.\n}\n$$X_{i}-X_{i-1}=\\begin{cases}\n                              \\mathbb{E}\\big[C_i(j)\\mid i\\in I_j, I_j\\cap [i-1]\\big]- \\mathbb{E}\\big[C_i(j)\\mid I_j\\cap [i-1]\\big], & \\mbox{if } i\\in I_j, \\\\\n                                        - \\mathbb{E}\\big[C_i(j)\\mid I_j\\cap [i-1]\\big] & \\mbox{otherwise}.\n                                       \\end{cases}$$\nTake any graph $R$ containing an edge of colour $i$. By the assumption, all edges of $R$ not ending in $U$ have colours in $[t,n]$, and therefore, $ \\mathbb{E}\\big[C_i(j) \\mid i\\in I_j, I_j\\cap [i-1]\\big]\\le C_i\\cdot p_j^{h-a}$ and $ \\mathbb{E}\\big[C_i(j)\\mid I_j\\cap [i-1]\\big]=p_j  \\mathbb{E}\\big[C_i(j)\\mid i\\in I_j, I_j\\cap [i-1]\\big]$. From here we may conclude that $|X_i-X_{i-1}|\\le C_i\\cdot p_j^{h-a}$ and, moreover, in terms of Theorem~\\ref{Azuma2}, $\\mathrm{Var}[X_i\\mid X_{i-1},\\ldots, X_1]=  \\mathbb{E}[|X_i-X_{i-1}|^2\\mid X_{i-1},\\ldots, X_1]\\le\\COMMENT{$p_j \\big((1-p_j)C_i p_j^{h-a}\\big)^2+(1-p_j)\\big(p_jC_i p_j^{h-a}\\big)^2=\\Big((1-p_j)^2+(1-p_j)p_j\\Big)C_i^2p_j^{2(h-a)+1}$} C_i^2 p_j^{2(h-a)+1}=:\\sigma_i^2.$\n\n\nNext, we have to bound $C_{i}$. Since $U$ is an independent set of $R$ for every $R \\in \\cF$, for any edge $uv$ of $R$, at least one of $u,v$ lies outside $U$. Moreover, if $u,v\\notin U$ then by our assumption $\\phi(uv)>t$. Hence, we obtain that $C_{i}$ equals the number of $R\\in \\cF$ which contain an edge $uv$ of colour $i$ which is incident to $U$.\\COMMENT{but not contained in}\nHence, $C_{i}$ is at most $f!n^{f-k-1}$ times the number of edges with colour $i$ in $G$ which are incident to $U$. Thus\n$C_{i} \\leq  f!k\\ell n^{f-k-1}$ for each $i\\in [t-1]$.\nMoreover, $\\sum_{i\\in [t-1]} C_i \\leq h|\\cF|$.\nIn terms of Theorem~\\ref{Azuma2}, this implies that\n$$\\sum_{i\\in [m]}\\sigma_i^2 \\le p_j^{2(h-a)+1}\\sum_{i=1}^{t-1}C_i^2+hf!n^{2f-2k-1\/2}\\le \\eps^{-1\/2} p_j^{2(h-a)+1}\\ell n^{2f-2k-1}. $$\nWe also have $$\\vartheta\\le \\max\\{n^{f-k-1\/2}, p_j^{h-a}\\cdot kf!\\ell n^{f-k-1}\\}\\le \\eps^{-1\/2}p_j^{h-a}\\ell n^{f-k-1}.$$\nSubstituting the right hand sides of the displayed formulas above in the inequality in Theorem~\\ref{Azuma2}, we have\n\\begin{multline*}\n\\mathbb{P}\\big[L \\ne \\big(1\\pm \\frac{\\eps}2\\big) p_j^{h}n^{f-k}  \\big] \\le 2\\exp\\Big(-\\frac{\\eps^{3} p_j^{2a-1}n^{2f-2k} }{\\ell n^{2f-2k-1}+\n\\eps p_j^{a-1}\\ell n^{2f-2k-1}}\\Big)\\le 2\\exp\\Big(-\\frac{\\eps^{4} p_j^{2a-1} n}{\\ell}\\Big).\\end{multline*}\nFinally,  the at most $n^{f-k-1\/3}$ potential  $R\\in \\cF$ that contain an edge of colour $i\\le t$, not incident to $U$, may change the value of $L$ by at most $\\frac{\\eps}2 p_j^h n^{f-k}$.\n\\end{proof}\n\n\n\\section{Approximate decompositions into near-spanning structures}\\label{sec5}\n\n\n\\subsection{Proof of Theorem~\\ref{thm: approx decomp}} The proof of this theorem is based on an application of Lemma~\\ref{lem: random F decomp}. It suffices to carry out some preprocessing and to verify that the conditions on the graph and the colouring are fulfilled.\n\nIf $h=0$, then there is nothing to prove, so we assume $h\\geq 1$.\nIf $f\\leq 2$ (and thus $F$ is an edge), then we replace $F$ by two disjoint edges, so we may assume that $f\\geq 3$.\n\n Choose $\\eta,\\zeta,\\eps, \\delta$ and $n_0$  such that  $0< 1\/n_0\\ll\\eta \\ll \\zeta\\ll \\eps \\ll \\delta\\ll \\alpha, d_0, 1\/f,1\/h$. Consider $G'$ as in the statement of the theorem. Let $V:=V(G')$ and  define $a,t\\in \\mathbb{N}$ by\n \\begin{equation}\\label{eqnumber} a:= \\frac{2h d^{h-1} n^{f-2}}{|\\Aut(F)|} \\enspace \\text{and} \\enspace t:=  \\frac{fdn}{2h}.\\end{equation}\n\nNote that $a \\geq \\eta n$ as $f\\geq 3$.\n\nBy Lemma~\\ref{qr to sr}, there is a $(\\zeta,d)$-quasirandom subgraph $G$ of $G'$ which satisfies \\eqref{qr 0}.\nLemma~\\ref{counting quasirandom} then implies that, for all $v\\in V$ and $e\\in E(G)$, we have\n\\begin{align}\\label{eq: two things}\nr_{G}(F,v) = (1\\pm \\eps\/3)t \\cdot a \\enspace \\text{and} \\enspace r_{G}(F,e) = (1\\pm \\eps\/3) a.\n\\end{align}\nWe claim that we can apply Lemma~\\ref{lem: random F decomp} with $\\cF$ being all rainbow copies of $F$ contained in $G$ and $\\alpha\/2$ playing the role of $\\delta$. Equations \\eqref{eqnumber} and \\eqref{eq: two things} imply that conditions \\ref{F decomp 3} and \\ref{F decomp 4} are satisfied (since $a\\ge \\eta n$). Condition~\\ref{F decomp 2} is satisfied since the number of copies of $F$ containing colour $i$ is at most\n$$e(G(\\phi,i))\\cdot \\max_{e\\in G} r_{G}(F,e)\\le (1+\\eta)\\frac{fdn}{2h}\\cdot (1+\\eps\/3)a\\le (1+\\eps\/2)ta.$$\n\nTo verify the codegree conditions \\ref{F decomp 1}, first note that for each $u \\neq v \\in V$, there are at most $f! n^{f-2}$ copies of $F$ containing both $u$ and $v$, so we have\n\\begin{align}\\label{eq: codegree codeg}\n|R_{G}(F,u)\\cap R_{G}(F,v)| \\leq f! n^{f-2}.\n\\end{align}\nFor $c\\neq c'\\in [m]$, we have\n\\begin{align}\\label{eq: codegree codeg 2}\n\\hspace{-1cm} &|\\{ \\bar F\\in R_{G}(F):  \\{c,c'\\} \\subseteq \\phi(E(\\Bar{F}))\\}| \\leq\n\\sum_{uv\\in E(G(\\phi,c))} (d_{G(\\phi,c')}(u)+d_{G(\\phi,c')}(v)) f! n^{f-3} \\nonumber\\\\\n&+ \\sum_{uv \\in E(G(\\phi,c'))} \\sum_{ u'v' \\in E(G(\\phi,c)-\\{u,v\\})} f! n^{f-4}\n\\leq fn \\cdot 2\\eta n \\cdot f! n^{f-3} + (fn)^2\\cdot f! n^{f-4} \\leq \\eta^{1\/2} n^{f-1}.\n\\end{align}\\COMMENT{We count copies of $F$ containing two edges $e,e'$ of colour $i$ and $i'$ such that either $|e\\cap e'|=1$ or $e\\cap e' =\\emptyset$. }\nSimilarly, one can obtain that for $c\\in [m]$ and $v\\in V$ one has \\begin{equation}\\label{eq: codegree codeg 3}R_{G}(F,v)\\cap\\{\\Bar{F}: c\\in \\phi(\\Bar{F})\\}\\le \\eta^{1\/2}n^{f-1}.\\end{equation}\n\\COMMENT{\\begin{align}\n\\hspace{-1cm} &|\\{ \\Bar{F}\\in R_{G}(F):  v\\in \\Bar{F},c \\in \\phi(E(\\Bar{F}))\\}| \\leq\n\\sum_{u\\in N_{G(\\phi,c)}(v)} f! n^{f-2} \\nonumber\\\\\n&+ \\sum_{uw \\in E(G(\\phi,c))} f! n^{f-3}\n\\leq \\eta n f! n^{f-2} + 2f! n^{f-2} \\leq 4f! \\eta n^{f-1}.\n\\end{align}}\nThus \\ref{F decomp 1} holds. Finally, for any two edges $e_1,e_2\\in E(G)$ we have $r_G(F,e_1\\cup e_2)\\le h^2n^{f-3}$ and thus \\ref{F decomp 5} is satisfied.\n\nTherefore, since $\\eta, \\eps\\ll\\alpha$ the conditions of Lemma~\\ref{lem: random F decomp} are satisfied, we obtain the desired $\\alpha$-decomposition into rainbow $\\alpha$-spanning $F$-factors.\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm: near spanning cycle}}\nLet us first present a sketch of the proof.\n\\begin{itemize}\n  \\item We start with splitting the graph into two smaller parts $V_1,V_2$ and one larger part $V_3$. Then we split the colours into a smaller part $I_1$ and a larger part $I_2$.\n  We make sure that most of the vertices and colours ``behave sufficiently nicely'': the graphs between the parts are $\\eps$-regular, the graphs inside the parts are quasirandom, and each colour appears roughly the ``expected'' number of times between and inside the parts (cf. Claim~\\ref{cla1}, \\eqref{prep -1} and \\eqref{prep 0}). We restrict our attention only to the colours and vertices that ``behave nicely''.\n  \\item Using Theorem~\\ref{thm: approx decomp}, we find an approximate decomposition of $V_3$ into rainbow almost-spanning factors consisting of long cycles using only the colours from $I_2$.\n  \\item For each cycle in each of these almost-spanning factors, we randomly select a ``special'' edge and remove it. The endpoints of these edges will be used to glue the cycles together into one long cycle. Again, we restrict our attention to cycles and colours that ``behave sufficiently nicely'': we discard all colours that appear ``unexpectedly'' many times between the endpoints of the ``special'' edges and the parts $V_1, V_2$, as well as all the cycles containing vertices of too high degree in these ``bad'' colours.\n  \\item Finally, we apply Lemma~\\ref{blowup} using ``good'' colours from $I_1$ to link up the endvertices of the  removed edges via $V_1$ and $V_2$. The fact that in the previous step we removed the ``special'' edges randomly guarantees us that we will be able to successively perform the connecting step for all the almost-spanning factors without causing the graph on $V_1,V_2$ and the colours in $I_1$ ``deteriorate'' too much during this process.\n\\end{itemize}\n\n\nLet us now make this precise. We choose auxiliary constants according to the hierarchy\n\\begin{small}\\begin{equation*} 1\/n_0\\ll\\eta\\ll \\zeta\\ll \\zeta_1\\ll\\epsilon\\ll\\eps_1\\ll 1\/s\\ll \\delta\\ll\\delta_1\\ll\\delta_2\\ll \\gamma\\ll \\beta\\ll\\alpha, d_0.\\end{equation*}\\end{small}\nTake a graph $G''$ and an $m$-colouring $\\phi$ of $G''$ satisfying the conditions of the theorem. Apply Lemma~\\ref{qr to sr} to $G''$ with  probability distribution $(q_1,q_2,q_3) = (\\delta_1, \\delta_1, 1-\\gamma)$, to obtain a $(\\zeta,d)$-quasirandom spanning subgraph $G'\\subseteq G''$, which, for any  $\\cV'':=(V''_1,V_2'',V''_3)$ chosen according to the above probability distribution, satisfies properties \\ref{qr 1}--\\ref{qr 4} with probability at least $0.9$.\n\nUsing Lemma~\\ref{lem: graph partition} for each colour $c$ and the graph $G'(\\phi,c)$,  we conclude that with probability at least $1-18e^{-\\zeta^3\\delta_1\/\\eta}\\ge 1-\\eta$ we have\n\\begin{align}\\notag e\\big(G'[V''_{1},V''_{2}](\\phi,c)\\big)\\le & 2\\delta_1^2 e(G'(\\phi,c))+\\zeta \\delta_1^2 n \\\\\n\\label{prep -1} \\le& 3\\delta_1^2 n.\\end{align}\nNote that the number of colours of size at least $3\\delta_1^2n$ in the {\\it original} colouring $\\phi$ is at most $e(G'')\/(3\\delta_1^2n)\\le \\eta^{-1\/3}n$. Using Markov's inequality, we also conclude that with probability at least $0.99$ at most an $\\eta^{2\/3}$-fraction of these does not satisfy \\eqref{prep -1}, so altogether \\eqref{prep -1} holds for all but  $\\eta^{1\/3}n$ colours. Similarly,\n\\begin{align} e\\big(G'[V''_3](\\phi,c)\\big)\\le& (1-\\gamma)^2e(G'(\\phi,c))+\\zeta^2 (1-\\gamma)^2 n\\notag\\\\\\le& \\label{prep 0}\\frac 12(1+\\zeta)(1-\\gamma)^2dn\\end{align}\nholds for all but $\\eta^{1\/3}n$ colours with probability $0.99$.\n\nChoose  $\\cV''=(V_1'',V_2'', V_3'')$ which satisfies conditions \\ref{qr 1}--\\ref{qr 4}, as well as \\eqref{prep -1}, \\eqref{prep 0} for all colours apart from a set $EC$ of at most $\\eta^{1\/4}n$ ``exceptional'' colours.\n\nLet $EV$ be the set of all those vertices $v$ with $d_{G'(\\phi,EC)}(v)\\ge \\zeta n$. Clearly, $|EV|\\le \\zeta n$. Put $G^*:=(G'[V_1'',V_2'',V_3'']\\setminus EV)-e(G'(\\phi,EC))$ and $V'_j:=V''_j\\setminus EV$ for $j\\in[3]$. By Proposition~\\ref{prop: edge deletion regular} (ii), we can find $V_j'\\subseteq V_j''\\setminus EV$ with $|V_j'|\\ge (1-\\zeta_1)|V_i''|$ such that\n \\begin{equation}\\label{eqstar} G:=G^*[V_1',V_2',V_3']\\text{ satisfies \\ref{qr 1}--\\ref{qr 4} with }V_j', \\zeta_1\\text{ playing the roles of }V_j,\\zeta,\\end{equation}\n as well as \\eqref{prep -1} and \\eqref{prep 0} for {\\it all} colours present in the colouring of $G$. We also note that $G$ has at least a $(1-\\gamma^{1\/2})$-fraction of vertices and edges of $G''$, therefore, an approximate decomposition into almost-spanning cycles for $G$ would be an approximate decomposition into almost-spanning cycles for the initial graph $G''$.\n\n\n\\begin{claim}\\label{cla1}\n A partition $\\cI:=(I_1,I_2)$ of the colours from $[m]\\setminus EC$, chosen at random with probability distribution $(p_1,p_2)=(\\gamma,1-\\gamma)$, with probability at least $0.9$ satisfies the following. There exist subsets $V_i\\subseteq V_i'$ for $i\\in [3]$, such that\n\\begin{enumerate}[label=\\text{{\\rm (V\\arabic*)$_{\\ref{cla1}}$}}]\n\\item \\label{prep 1} $|V_i|\\ge (1-\\zeta_1)|V_i'|$.\n\\item \\label{prep 2} The graph $G[V_1,V_2](\\phi, I_1)$ is $(\\eps^{1\/6},\\gamma d)$-regular.\n\\item \\label{prep 3} For each vertex $v\\in V_3$ and $i\\in[2]$ we have $d_{G(\\phi, I_1),V_i}(v)=(1\\pm \\eps)\\gamma d |V_i|$.\n\\item \\label{prep 4} The graph $G[V_3](\\phi, I_2)$ is $(\\eps, (1-\\gamma)d)$-quasirandom.\n\\end{enumerate}\n\\end{claim}\n\\begin{proof}\nConsider a partition $\\cI:=(I_1,I_2)$  of the colours  as in the claim. Apply Lemma~\\ref{lem: colour partition} with partition $\\cI$ for each vertex $v\\in V(G)$, where $\\eps^2$ plays the role of $\\eps$,  $U:=\\{v\\}$ and $\\cF$ is simply the collection of all edges in $G$ from $v$ to $V_i'$. Keeping in mind  that, by \\eqref{eqstar}, $G$ is $(\\zeta_1,d)$-superregular between the parts and $(\\zeta_1,d)$-quasirandom inside the parts, we conclude that  for all $j\\in[2]$ and $i\\in[3]$ we have \\begin{equation}\\label{prep5} d_{G(\\phi,I_j),V_i'}(v)=(1\\pm \\eps\/2)p_j d |V_i'|\\end{equation}\n with probability at least $1-12\\exp(-\\frac{\\eps^8\\gamma}{\\eta})>1-\\eta$. Using Markov's inequality, with probability at least $0.99$ the number of vertices not satisfying \\eqref{prep5} is at most $\\eta^{1\/2}n$. Delete these vertices, obtaining sets $V_i\\subseteq V_i'$, $i\\in [3]$. Note that they satisfy \\ref{prep 1}, and that the condition \\ref{prep 3} is fulfilled as well. (Indeed, $d_{G(\\phi,I_j),V_i}(v)=(1\\pm \\eps\/2)p_j d |V_i'|\\pm\\eta^{1\/2}n=(1\\pm \\eps)p_j d |V_i|$.)\\COMMENT{We use this later to prove\\ref{prep 2}.}\n\n\n  Fix $i_1,i_2\\in[3]$. Since $G$ satisfies \\ref{qr 2} with $\\zeta_1$ playing the role of $\\zeta$, it follows  that the total number of pairs of vertices $u,v\\in V_{i_1}'$, for which $d_{G,V_{i_2}'}(u,v)\\ne (d^2\\pm \\zeta_1)|V_{i_2}'|$ is at most $\\zeta_1 n^2$. Moreover, the total number of pairs $u,v$, which have more than $\\eta^{1\/2}n$  monochromatic paths $P_2$ with ends in $v$ and $u$ is at most $\\eta^{1\/2}n^2$ by Lemma~\\ref{lem: colour irregular degree}.\\COMMENT{(Note that the number of all such $P_2$, not necessarily monochromatic, is the codegree of $u$ and $v$.)}\n \n  Consider any pair of vertices $u,v\\in V_{i_1}$ which does not belong to either of these two sets $EP_1$ and $EP_2$ of ``exceptional'' pairs. Then we conclude that the number of rainbow (i.e. two-coloured) $P_2$ with ends in $u$ and $v$ and  middle vertex in $V_{i_2}$ is $(d^2\\pm 3\\zeta_1)|V_{i_2}|$. Here we both used that $|V_{i_2}|\\ge (1-\\zeta_1)|V_{i_2}'|$ and that all but $\\eta^{1\/2}n$ copies of $P_2$ with ends in $v,w$ are rainbow. Apply Lemma~\\ref{lem: colour partition} with $U:=\\{u,v\\}$, $\\eps^2$ playing the role of $\\eps$ and $\\cF$ being a collection of rainbow $P_2$ with ends in $u$ and $v$ and  middle vertex in $V_{i_2}$. We conclude that for each $j\\in [2]$ the number of rainbow $P_2$ which end in $u,v$, have their middle vertex in $V_{i_2}$ and are coloured with colours from $I_j$ is\n\\begin{equation}\\label{prep6} (1\\pm \\eps\/3)(d^2\\pm 3\\zeta_1)p_j^2|V_{i_2}|=(1\\pm\\eps\/2)d^2p_j^2|V_{i_2}|\\end{equation}\n with probability $1-4\\exp(-\\frac{\\eps^8\\gamma^3}{\\eta})>1-\\eta$. Using Markov's inequality, with probability $0.99$ the number of pairs $(u,v)\\notin EP_1\\cup EP_2$ violating \\eqref{prep6} is at most $\\eta^{1\/2}n^2$. For any pair $u,v\\in V_{i_1}$ not belonging to the set $EP_3$ of these ``exceptional'' pairs we have\\COMMENT{note that we also have to account for the monochromatic codegree of two vertices, which is at most $\\eta^{1\/2}n$.}\n \\begin{equation}\\label{prep7} d_{G(\\phi, I_j),V_{i_2}}(u,v)=(1\\pm \\eps\/2)d^2p_j^2|V_{i_2}|\\pm \\eta^{1\/2}n=(1\\pm \\eps)d^2p_j^2|V_{i_2}|.\\end{equation}\n Proceed in a similar way for all choices of $i_1,i_2\\in [3]$. Then the union of the sets $EP_1\\cup EP_2\\cup EP_3$ of all exceptional pairs (taken over all choices of $i_1,i_2\\in[3]$) has size at most $9\\zeta_1n^2+9\\eta^{1\/2}n^2+9\\eta^{1\/2}n^2\\le \\eps^{2}\\delta_1^2n^2$.\nIn particular, we may conclude that all but at most an $\\eps$-proportion of pairs in $V_3$ satisfy \\eqref{prep7} with  $j=2$ and $i_2=3$. Together with \\eqref{prep5} this implies that $G[V_3](\\phi, I_2)$ is $(\\eps, (1-\\gamma)d)$-quasirandom, i.e., \\ref{prep 4} holds. Similarly, by Theorem~\\ref{thm: almost quasirandom}, property~\\ref{prep 2} is satisfied.\n \\end{proof}\n\n\nAfter this preprocessing step, we are ready to proceed with the construction of our almost-decomposition. First, apply Theorem~\\ref{thm: approx decomp} to $G[V_3](\\phi,I_2)$ for $F:=C_s$  and with $3\\eps, \\beta$ playing the roles of $\\eta, \\alpha$ (recall that $\\eps\\ll1\/s\\ll\\delta\\ll \\beta $). Indeed, to see that we can apply Theorem~\\ref{thm: approx decomp}, first note that the colouring on $G[V_3](\\phi,I_2)$ is locally $\\eps |V_3|$-bounded since $\\eta n\\le \\eps |V_3|$. Moreover, due to \\eqref{prep 0}, it is $\\frac 12 (1+\\zeta)(1-\\gamma)^2dn\\le \\frac 12(1+\\eps)(1-\\gamma)d|V_3|$-bounded.\\COMMENT{Note that the average density in $G[V_3](\\phi, I_2)$ is  $\\ge (1-\\eps)(1-\\gamma)d$.} As a result, we obtain a $\\beta$-decomposition of $G[V_3](\\phi,I_2)$ into rainbow $\\beta$-spanning $C_s$-factors. Denote by $\\cL_i'$ the $i$-th factor from this decomposition, and let $n_1$ be their total number. By deleting some cycles if necessary, we may assume that each factor includes the same number $n_2'$ of copies of $C_s$, where $n_2'\\ge (1-2\\beta)\\frac ns$. That is, $\\cL_i':=\\bigcup_{j=1}^{n_2'} C_i^j$, where $C_i^j$ are the $s$-cycles forming $\\cL_i'$.\n Thus \\begin{equation}\\label{eqspan1}\n        |V(\\cL_i')|\\ge (1-2\\beta)n \\ \\ \\ \\ \\ \\text{for each }i\\in [n_1].\n      \\end{equation}\n      The union of all the  $\\cL_i'$ covers all but a $4\\beta$-fraction of the edges of the initial graph $G''$.\n\nThe last step of the proof is  to combine (most of) the cycles in each $\\cL_i'$ into one large cycle using the vertices from $V_1$, $V_2$ and the colours from $I_1$. For all $i\\in[n_1]$ and $j\\in [n_2']$ select an edge $e_i^j=x_i^jy_i^j$ in $C_i^j$ independently uniformly at random. Put $U'_i:=\\bigcup_{j=1}^{n_2'}\\{x_i^j,y_i^j\\}$. We claim that the following two properties have non-zero probability to be satisfied simultaneously:\n\\begin{itemize}\n  \\item[\\textbf A] Each vertex $v\\in V_3$ belongs to at most $\\delta n$ selected edges.\n  \\item[\\textbf B] For each $i\\in [n_1]$ define $I^i$ to be the set of colours $c\\in I_1$ such that $e(G[V_1\\cup V_2, U_i'](\\phi, c))>\\delta n$. Then $e(G[V_1\\cup V_2, U_i'](\\phi, I^i))\\le \\delta e(G[V_1\\cup V_2, U_i'](\\phi, I_1))$, as well as $e(G(\\phi, I^i))\\le \\delta n^2$.\n\\end{itemize}\nLet us verify this claim. For each $i\\in[n_1]$, any given $v\\in V_3$ belongs to at most one $C_i^j$, and thus it belongs to the corresponding $e_i^j$ with probability at most $2\/s$. Using Lemma~\\ref{Chernoff} and a union bound, the probability that the property \\textbf{A} does not hold is at most $2n e^{-\\delta n}\\le e^{-\\delta n\/2}.$\n\n\n\n Since $e(G[V_1\\cup V_2, U_i'])\\le \\frac{5\\delta_1 n^2}s$ by \\ref{qr 1}, the definition of $I^i$ implies that $|I^i|\\le \\delta n$, and so in particular $e(G(\\phi,I^i))\\le \\delta n^2$. For any fixed $i\\in[n_1]$ and a colour $c\\in I_1$, the expected value of $e(G[V_1\\cup V_2, U_i'](\\phi, c))$ is at most $2n\/s$, and so by Markov's inequality, $c\\in I^i$ with probability at most $\\frac{2}{\\delta s}<\\delta^3$. Therefore, the expected number of edges in $G[V_1\\cup V_2,U_i']$ having colours from $I^i$ is at most $\\delta^3 e(G[V_1\\cup V_2, U_i'](\\phi, I_1))$. Using  Markov's inequality again, with probability at least  $1-\\delta^2$ the number of such edges is at most\n$\\delta e(G[V_1\\cup V_2, U_i'](\\phi, I_1))$.\nCombining this bound for different values $i$, we obtain that property \\textbf{B} is satisfied with probability at least $(1-\\delta^2)^{n_1}>e^{-\\delta n\/3}$.\\COMMENT{We use that $n_1\\le n$ and that for any $\\alpha<\\beta$ there exists $x_0>0$, such that for any $0<x<x_0$ we have $(1-x)^{\\alpha}>e^{-\\beta x}$.} Therefore, with positive probability both \\textbf{A} and \\textbf{B} are satisfied. Fix a choice of edges satisfying both \\textbf A and \\textbf B simultaneously.\n\nFor each $i\\in[n_1]$, define $J_i\\subseteq[n_2']$ to be the set of indices such that $j\\in J_i$ if and only if at least a $\\delta_1$-proportion of edges in $G[V_1\\cup V_2,z](\\phi,I_1)$ are coloured in colours from $I^i$ for at least one $z\\in \\{x_i^j,y_i^j\\}$.\nDue to \\textbf B and \\ref{prep 3}, we have \\begin{equation}\\label{eqspan2}\n                                             |J_i|\\le \\delta_1 n_2'.\n                                           \\end{equation}\nBy \\eqref{eqspan1} we have \\begin{equation}\\label{eqspan3}\n                             \\Big| \\bigcup_{j\\in[n_2']\\setminus J_i} V(C_i^j)\\Big|\\ge (1-3\\beta)n.\n                           \\end{equation}\nBy disregarding some cycles if necessary, we assume that the $J_i$ have the same cardinality for any $i$ and that the cycles are ordered in such a way that $[n_2']\\setminus J_i = [n_2]$ for some $n_2<n_2'$. Put $U_i:= \\bigcup_{j=1}^{n_2}\\{x_i^j,y_i^j\\}$. Then the following holds.\n\\begin{itemize}\n  \\item[\\textbf C] For all $i\\in [n_1]$, $z\\in U_i$ and $q\\in [2]$ we have $d_{G(\\phi,I^i),V_q}(z)\\le \\delta_1 d_{G(\\phi,I_1),V_1\\cup V_2}(z)\\overset{\\ref{prep 3}}{\\le} 5\\delta_1\\gamma d|V_q|$.\n\\end{itemize}\n\nFinally, we are ready to apply Lemma~\\ref{blowup}. For each $i\\in [n_1]$ in turn, we define a mapping $\\psi_i$ as follows. Consider the graph $H$ consisting of $n_2$ vertex-disjoint copies of $P_3$, say, $H:=\\bigcup_{j=1}^{n_2}P_3^j$, where   $P_3^j:=w_0^jw_1^jw_2^jw_3^j$ is a path of length $3$. Consider the partition $\\cX =(X_0,X_1,X_2)$ of $V(H)$, where $X_q :=\\{w_q^j: j\\in [n_2]\\}$ for $q=1,2$, and $X_0:=\\{w_0^j,w_3^j: j\\in [n_2]\\}.$ Consider the map $\\psi'_i: X_0\\to U_i$, defined by $\\psi'_i(w_0^j)=x_i^j$ and $\\psi'_i(w_3^j)=y_i^{j+1}$, with indices taken modulo $n_2$. Assume that we have already defined $\\psi_1,\\dots, \\psi_{i-1}$.\nThen, provided that the necessary conditions hold, Lemma~\\ref{blowup} will guarantee\n$$\\text{an embedding  }\\psi_i\\text{ of }H\\text{ into }G_i:=G[V_1,V_2,U_i](\\phi, I_1\\setminus I^i)-\\bigcup_{k=1}^{i-1} E(\\psi_{k}(H)),$$\nextending $\\psi'_i$, such that $\\psi_i(X_q)\\subseteq V_q$ for $q\\in[2]$ and $\\psi_i(H)$ is rainbow. By \\eqref{eqspan3}, for each $i\\in [n_1]$, the union of $\\psi_i(H)$ and all the  $C_i^j- e_i^j$ with $j\\in [n_2]$ gives us a rainbow cycle of length at least $(1-3\\beta)n$. These cycles are edge-disjoint and altogether provide us with an $\\alpha$-decomposition of $G''$ (by \\eqref{eqspan2} and since the $\\cL_i'$ form a $4\\beta$-decomposition of $G''$). Thus, it only remains to verify the conditions of Lemma~\\ref{blowup} in order to complete the proof of the theorem. (For this, we will make use of \\eqref{eqstar}.)\nWe apply Lemma~\\ref{blowup} with $G_i$, $\\delta_1 n$ playing the roles of $G, n$.\n\\begin{itemize}\\item Due to \\eqref{prep -1} and the definition of $I^i$ in \\textbf{B}, the colouring of $G_i$ is $3\\delta_1^2n+\\delta n\\le \\delta_2 \\delta_1n$-bounded.\n\\item\\ref{lem blowup 1} is clear from the definition of $H$, with $\\Delta=2$.\n\\item\\ref{lem blowup 2} is satisfied since $|X_0|=|U_i|\\le 2n\/s\\le \\delta_2\\delta_1 n$.\n\\item \\ref{lem blowup 3} is satisfied in the form needed to apply the ``moreover part'' of Lemma~\\ref{blowup} since $|X_j|=\\frac 12 |U_i|<\\delta_2 |V_j|$ for $j\\in[2]$ and $$|V_j|\\overset{\\ref{prep 1}}{=}(1\\pm \\zeta_1)|V_j'|\\overset{\\eqref{eqstar}}{=}(1\\pm\\zeta_1)(1\\pm\\zeta_1)\\delta_1n=(1\\pm\\delta_2)\\delta_1 n.$$\n\\item Due to $e(H)\\le 3n\/s\\le \\delta n$ and property \\textbf B, the graph $\\bigcup_{k=1}^{i-1}\\psi_{k}(H)\\cup G(\\phi, I^i)$ contains at most $2\\delta n^2$\n    edges.\n    Assertion \\ref{prep 2} together with Proposition~\\ref{prop: edge deletion regular} imply that $G_i[V_1,V_2]$ is $(\\delta_2, \\gamma d)$-regular  (note here that $\\delta_2\\ll \\gamma$). Therefore, \\ref{lem blowup 4} is satisfied with the prefix ``super'' omitted, which is sufficient to apply the ``moreover part'' of Lemma~\\ref{blowup}.\n\\item Finally, \\ref{lem blowup 5} is satisfied due to \\ref{prep 3}, combined with \\textbf{A} and \\textbf{C}.\n\\end{itemize}\n\n\n\n\n\n\n\n\n\\section{Spanning structures}\\label{sec6}\n\n\\subsection{Proof of Theorem~\\ref{thm: perfect decomp}. Almost-decomposition into $F$-factors}\n\nLet us first give a sketch of the proof.\n\n\\begin{itemize}\n  \\item We split the vertex set $V$ of $G$ into $b=O(\\log n)$ equal size parts $U_i$, and, using Lemma~\\ref{qr to sr}, we have that the pairs of parts induce superregular pairs. We ignore the edges inside each $U_i$ since the number of these is negligible.\n  \\item Using the result on resolvable designs (Theorem~\\ref{cl: r-factor}), we split the collection of $U_i$ into groups of $f$ parts, such that each pair of parts belongs to exactly one group, and the set of all groups has a partition into layers with each layer covering all $U_i$ exactly once. In other words, we consider a decomposition of the complete graph $K_b$ into $K_f$-factors. Next, we aim to translate this into an almost-decomposition of $G$. Each $K_f$ from the decomposition will correspond to an $f$-partite graph $G_i^j\\subseteq G$ between the corresponding parts.\n  \\item  We randomly split the colours as follows. We set aside a small proportion of colours $I_{q+1}$, and the remaining ones we split into $q:=b\/f$ groups of roughly equal size. By Lemma~\\ref{lem: colour partition}, in each of the colour groups and for each of the $G_i^j$ the count of rainbow copies of $F$ is ``correct''. Thus, we can apply Lemma~\\ref{lem: random F decomp} and obtain an approximate decomposition into rainbow almost-spanning $F$-factors of each of $G_i^j$ in each of the colour groups.\n   \\item Next we combine the rainbow almost-spanning $F$-factors in all the $G_i^j$ into rainbow almost-spanning $F$-factors in $G$.\n   \\item Finally, we transform each such almost-spanning $F$-factor $\\cD$ into an $F$-factor as follows. Let $V_2$ be the set of vertices not covered by $\\cD$. Since we have little control over $V_2$, we also consider a set of vertices $V_1$ (depending on $\\cD$) which is the union of several $V(G_i^j)$, where these $i,j$ are chosen in a way that, over all $F$-factors, each $V(G_i^j)$ is used roughly the same number of times by the sets $V_1$. Moreover, we will have $|V_2| \\ll |V_1| \\ll n$. We discard $\\cD[V_1]$ and then apply the rainbow blow-up lemma to obtain a rainbow spanning factor on $V_1\\cup V_2$ in colours from $I_{q+1}$. Combined with $\\cD[V\\setminus(V_1\\cup V_2)]$,  this gives a rainbow  $F$-factor in $G$. Altogether, these $F$-factors form the desired approximate decomposition.\n\n\\end{itemize}\n\nThe main challenge in the final step is to carry this out in such a way that the conditions of the rainbow blow-up lemma~(Theorem~\\ref{blowup}) are satisfied in each successive application. In particular, we need to show that the colouring is well-bounded and that in each iteration, the vertex degrees are not affected too much.\nThis is the main reason why we need the almost-spanning factors to be distributed randomly in the assertion of Lemma~\\ref{lem: random F decomp}. This guarantees that, when we combine almost-spanning  factors from different $G_i^j$, the vertices that are left out will ``behave nicely'' with respect to each colour in $I_{q+1}$, in particular, the edges of each colour from $I_{q+1}$ will appear roughly the correct number of times. The degrees of the vertices are not affected too much since none is used too many times in some $V_1$ and since the random choice of the approximate decompositions of $G_i^j$ does not allow a vertex to appear too many times in some $V_2$.\n\n\nLet us make this precise. If $h=0$, then there is nothing to prove, so we assume $h\\geq 1$.\nIf $f\\leq 2$ (and thus $F$ is an edge), then we replace $F$ with two disjoint edges, so we may assume that $f\\geq 3$. We remark that this does not change the value of $a$ from the statement. However, this affects the divisibility conditions  (if $n$ is divisible by $2$, but not by $4$), but this problem is easy to fix, and we will come back to it when applying the rainbow blow-up lemma.\n\nWe choose auxiliary constants according to the hierarchy \\begin{small}\\begin{equation}\\label{eqhier} 0<1\/n_0\\ll\\eta\\ll \\zeta\\ll \\zeta_1\\ll\\epsilon\\ll \\delta\\ll\\delta_1\\ll\\delta_2\\ll \\gamma\\ll\\alpha, d_0 \\ \\ \\ \\text{and} \\ \\ \\ \\delta_2\\ll 1\/f, 1\/h. \\end{equation}\\end{small}\n\nLet $G'$ be a graph with an $m$-colouring $\\phi$ as in the formulation of Theorem~\\ref{thm: perfect decomp}. Let us note that for any $c\\in[m]$\n\\begin{equation}\\label{colorbound} e(G'(\\phi,c))\\le \\frac f{h} n.\n\\end{equation}\nWithout loss of generality, assume that the number $b':=\\eta^{-1\/3a}\\log n$ is an integer, and define integers\n$$b:=f(f-1)b'+f,\\ \\ \\ \\ \\ g:=(b-1)\/(f-1),\\ \\ \\ \\ \\ q:=b\/f.$$\nApply Lemma~\\ref{qr to sr} to $G'$ to obtain a $(\\zeta,d)$-quasirandom spanning subgraph $G$ of $G'$ such that a random partition $\\cU:=(U_1,\\ldots, U_b)$ of $V(G')$ chosen with probability distribution $(1\/b,\\ldots, 1\/b)$ satisfies the following with probability at least 0.9.\n\n \\begin{enumerate}[label=\\text{{\\rm (U\\arabic*)}}]\n\\item \\label{U1} For each $i\\in [b]$, we have $|U_i|= (1\\pm \\zeta)n\/b$.\n\\item \\label{U2} For all $i\\neq j \\in [b]$, the bipartite graph $G[U_{i},U_{j}]$ is $(\\zeta,d)$-superregular.\n\\item \\label{U3} For all $vw\\in E(G)$ and $i\\in [b]$, we have $|C_G^{\\phi}(v,w)\\cap U_i|\\le \\zeta |U_i|$ and\n$d_{G,U_{i}}(v,w) = (d^2 \\pm \\zeta)|U_{i}|$.\n\\end{enumerate}\nA Chernoff estimate also shows that the following holds with probability at least $0.9$.\n\\COMMENT{\n Colour degree $X$ satisfies $\\mathbb{E}[X]\\leq \\eta n\/(b \\log^{2a}n) := s$. $\\mathbb{P}[X\\geq (1+2\\zeta)s] \\leq 2e^{- (\\zeta s)^2\/(2\\cdot 4s\/3)}.$}\n\\begin{enumerate}[label=\\text{{\\rm (U\\arabic*)}}]\n  \\setcounter{enumi}{3}\n\\item\\label{U4} For each $i,j\\in[b]$ the colouring $\\phi$ of $G[U_i,U_j]$ is locally $(1+2\\zeta)\\eta \\frac{n\/b}{\\log^{2a} n}$-bounded.\\COMMENT{We take a union bound over at most $n^4$ events, where events are ``the intersection of a big set and a randomly chosen set are roughly of correct size''.}\n\\end{enumerate}\n\n\nNext apply Lemma~\\ref{lem: graph partition} to $G(\\phi,c)$ with a random partition $\\cU$ as above and with $\\eta^{1\/8}$ playing the role of $\\zeta$ to obtain that\n\\begin{equation}\\label{eqcolor} e(G[U_i,U_j](\\phi,c))=\\frac 2{b^2}e(G(\\phi,c))\\pm \\frac{\\zeta}{b^2}n\\end{equation}\nholds with probability at least $0.9$ for every $c\\in [m]$ and $i,j\\in [b]$.\\COMMENT{The probability that \\eqref{eqcolor} is satisfied for all colours and all $i,j$ is at least $$1-2n^2(\\log{n})^2e^{-\\frac{\\eta^{3\/8}n\/b}{\\eta n\/\\log^2 n}}\\ge 1-n^3 e^{-\\eta^{-1\/8}\\log n}=1-o(1).$$\n(We can take $\\ell = \\eta n \\log^{-2}n$, $p_i=1\/b$ and $\\zeta=\\eta^{1\/8}$.)\n}\n Fix one such partition $\\cU$ satisfying \\ref{U1}--\\ref{U4} as well as \\eqref{eqcolor}.\n\nDue to the choice of $b$, we can apply Theorem~\\ref{cl: r-factor} with $f,b'$ playing the roles of $r,b'$ and $\\rho=1$. Let $\\cL_1,\\ldots, \\cL_{g}$ be the perfect $f$-matchings on $[b]$ thus obtained, and, for each $i\\in [g]$, write $\\cL_i=\\{L_i^j:j\\in[q]\\}$.\n\nFor each $f$-tuple $L_i^j=:\\{i'_1,\\ldots, i'_f\\}$ with $i\\in [g]$ and $j\\in [q]$, let $G_i^j:= G[U_{i'_1},\\ldots, U_{i'_f}]$.  Next we apply Lemma~\\ref{counting partite} with $G_i^j$ playing the role of $G$ (note that the assertions \\ref{U1}, \\ref{U2}, \\ref{U3} immediately imply the conditions \\ref{lem counting partite 2}, \\ref{lem counting partite 1},  \\ref{lem counting partite 3}, respectively, and \\eqref{eqcolor} guarantees the required boundedness of the colouring, while \\ref{U4} implies the local boundedness of the colouring).  We apply Lemma~\\ref{counting partite}  to $F$ with the ``trivial'' partition of $V(F)$ into parts of size $1$ and to $G_i^j$ with its natural partition (these are the only partitions we use in what follows, so, by abuse of notation, we will not specify them in the notation). We obtain that for each $v\\in V(G_i^j)$ and $uw\\in E(G_i^j)$\n\\begin{eqnarray}\\label{corr count 1} r_{G_i^j}(F,v) &\\ge & \\frac 12d^h (n\/b)^{f-1}\\enspace \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{ and } \\\\\n\\label{corr count 2} \\frac{r_{G_i^j}(F,v)}{r_{G_i^j}(F,uw)} &= & (1\\pm 2\\eps)\\frac{ {f\\choose 2}dn}{bh}=(1\\pm 3\\eps)\\frac{f|E(G_i^j)|}{h|V(G_i^j)|}.\n\\end{eqnarray}\nThe final equality holds since  \\ref{U1} and \\ref{U2} imply that the average degree in $G_i^j$ is $(1\\pm 3\\zeta)\\frac{dn(f-1)}{b}$.\n\nConsider a random partition $\\cI:=(I_1,\\ldots, I_{q+1})$ of colours chosen with  probability distribution $(p_1,\\ldots, p_{q+1}):=(\\frac {1-\\gamma}q,\\ldots, \\frac{1-\\gamma}q, \\gamma)$. Note that the number of colour classes, excluding the last one, is equal to the number of $f$-tuples in each $\\cL_i$. For all $i\\in[g], j\\in[q]$, apply Lemma~\\ref{lem: colour partition} to $G_i^j$ with $\\emptyset$, $\\eps^2, |V(G_i^j)|,E(G_i^j)$ playing the roles of $U,\\eps,n,\\cF$. Together with \\ref{U4} this implies that with probability $1-o(1)\n\\COMMENT{By \\ref{U4} we can take $\\ell = 2\\eta \\frac{n\/b}{\\log^{2a}n}$.\nRecall $q^{-2a-1} \\sim \\eta^{\\frac{2a-1}{3a}} \/ (\\log n)^{2a-1}$.\nThen \\eqref{corr count 6} holds with probability $\\ge 1-2q^2ge^{-\\frac{\\eps^9n\/q}{2\\eta n\/(b\\log ^{2a}n) q^{2a-1}}}\\ge 1-2q^2ge^{-\\eta^{-1\/3}\\log n}.$    } for all $j,r\\in[q]$, $i\\in [g]$ we have\n\\begin{equation}\\label{corr count 6} |E(G_i^j(\\phi, I_r))|=(1\\pm \\eps\/2)p_r|E(G_i^j)|.\n\\end{equation}\nNext, for each $i\\in[g]$, $j\\in[q]$ and\n$v\\in V(G_i^j)$, we apply  Lemma~\\ref{lem: colour partition} to $G_i^j$ with $\\{v\\},\\eps^2,\\cF$ playing the roles of $U,\\eps,R_{G_i^j}(F,v)$. Next, for each $uw\\in E(G_i^j)$ we apply the lemma with $\\{u,w\\},\\eps^2,\\{F'-uw:F'\\in R_{G_i^j}(F,uw)\\}$ playing the roles of $U,\\eps,\\cF$. Finally, for any $u'w_1,u'w_2\\in E(G_i^j)$ we apply the lemma with $\\{u',w_1,w_2\\},\\eps,\\{F'-u'w_1-u'w_2-w_1w_2:F'\\in R_{G_i^j}(F,u'w_1\\cup u'w_2)\\}$ playing the roles of $U,\\eps,\\cF$.\n We claim that with probability $1-n^{-1}$ for all $i\\in [g],\\ j\\in[q]$, $r\\in [q]$, all vertices\n$v\\in V(G_i^j)$ and edges $uw,u'w_1,u'w_2\\in E(G_i^j(\\phi,I_r))$ we have\n\\begin{eqnarray}\\label{corr count 3} r_{G_i^j(\\phi,I_r)}(F,v) &\\ge& \\frac 14d^hp_{r}^h (n\/b)^{f-1}, \\\\\n\\label{corr count 4} \\frac{r_{G_i^j(\\phi,I_r)}(F,v)}{r_{G_i^j(\\phi,I_r)}(F,uw)} &=& (1\\pm4\\eps)\\frac{p_rf|E(G_i^j)|}{h|V(G_i^j)\\big|}\\overset{\\eqref{corr count 6}}{=}(1\\pm 5\\eps)\\frac{f|E(G_i^j(\\phi, I_r)|}{h|V(G_i^j)|},\\\\\n\\label{corr count 7} r_{G_i^j(\\phi,I_r)}(F,u'w_1\\cup u'w_2) &\\le& 2f!p_{r}^{h-3} (n\/q)^{f-3}.\n\\end{eqnarray}\nIndeed, to see \\eqref{corr count 3} and \\eqref{corr count 4}, note that in \\eqref{corr count 3} we combined \\eqref{corr count 1} with the conclusion of Lemma~\\ref{lem: colour partition}, while in \\eqref{corr count 4} we combined \\eqref{corr count 2} and the conclusions of Lemma~\\ref{lem: colour partition} (obtained from fixing $v$ and then $uw$).  To see \\eqref{corr count 7}, we first use the trivial bound $r_{G_i^j}(F,u'w_1\\cup u'w_2) \\le \\frac 32 f!(n\/q)^{f-3}$ and then apply Lemma~\\ref{lem: colour partition}. Let us check that a union bound allows us to arrive to the desired conclusion. First note that the maximum number of edges incident to $U$ in the applications of Lemma~\\ref{lem: colour partition} is bounded by $a$.\\COMMENT{ Indeed, we have $U$ of three types: either it is a vertex of $F$ or it is a two-element independent set, formed after removing an edge of $F$, or, finally, it is a three-element set, formed after removing a path of length two. (or a triangle $uvw$, but then the number of edges\nincident is $d(u)+d(v)+d(w)-6 \\le a''(F)$)}\nUsing \\ref{U4}, the probability that \\eqref{corr count 3}--\\eqref{corr count 7} hold for fixed $i,j,r,v, u'w_1,u'w_2$ and $uw$ is at least\n$$\n1-2\\exp\\Big(-\\frac{\\eps^9p_r^{2a-1}\\frac{n}{q}}{2\\eta \\frac nb\\log^{-2a}n}\\Big)\\ge 1-\\exp\\big(-\\eta^{-1\/3}\\log n\\big)\\ge 1-n^{-10}.\n$$\nThus, taking a union bound over all possible choices of $i,j,r$ and $v$ as well as $uv,u'w_1,u'w_2$, we conclude that \\eqref{corr count 3}--\\eqref{corr count 7} hold for all such choices simultaneously with probability at least $1-n^{-1}$.\n\nMoreover, adapting the proof of \\ref{prep 2} to our setting, we have\\COMMENT{Could get $\\epsilon$ instead of $\\epsilon^{1\/6}$ but $\\epsilon^{1\/6}$ makes it more similar to \\ref{prep 2}}\n\\begin{equation}\\label{corr count 5} G[U_i,U_j](\\phi, I_{q+1})\\ \\ \\ \\ \\text{ is }(\\eps^{1\/6}, \\gamma d)\\text{-superregular for any }i\\ne j\\in [b]\n\\end{equation}\nwith probability $1-n^{-1}$.\\COMMENT{ By \\ref{U2}, $G[U_i,U_j]$ is $(\\zeta,d)$-superregular.  Applying Lemma~\\ref{lem: colour partition} to $G[U_i,U_j]$ for each $v\\in U_i\\cup U_j$ and $\\cF$ being the set of edges incident to $v$ in $G[U_i,U_j]$, it is clear that all vertices w.h.p. have correct degree. To show that the graph is actually $(\\eps,\\gamma d)$-regular, we show that most vertices in each pair have correct codegree and then apply Theorem~\\ref{thm: almost quasirandom}. Applying Lemma~\\ref{lem: colour irregular degree} to $G$ with $\\ell=\\eta n\/\\log^2n$ (this is the maximum value of $\\ell$ we can have as the upper bound on the local-boundedness), and $|U_i\\cup U_j|$ playing the role of $n$ we conclude that each vertex in $U_i$ has at most $\\eta^{1\/2} n^{1\/2}|U_i\\cup U_j|^{1\/2}\/\\log n\\le \\eta |U_i|$ other vertices in $U_i$ with which it has monochromatic codegree bigger than $\\eta|U_i|$. Ignore all pairs that have big monochromatic codegree (they form the set $EP_2$ in terms of Claim~\\ref{cla1}, which satisfies $|EP_2|\\le \\eps^2|U_i|^2$).\nWe also ignore the set $EP_1$ of pairs of codegree not close to $d^2|U_i|$.\nAs for the others, their codegree in $U_j$ is largely formed by the rainbow paths $P_2$ ending in $U_j$ (all but a $\\eta^{1\/3}$-fraction of the codegree). Apply  Lemma~\\ref{lem: colour partition} with $\\cF$ being rainbow paths of length $2$ between a given such pair $u,v$ of vertices  and ending in $U_j$ and conclude that with high probability their ``rainbow'' codegree of $u,v$ in $G[U_i,U_j](\\phi, I_{q+1})$ is roughly $\\gamma^2$ smaller than their ``rainbow'' codegree in $G[U_i,U_j]$. Adding back the monochromatic codegree only slightly affects the error term since $\\eta^{1\/3}\\ll \\gamma^2$. Thus, by Theorem~\\ref{thm: almost quasirandom} the graph is $(\\eps^{1\/6},\\gamma d)$-regular with (very) high probability.} From now on, we fix a colour partition $\\cI=(I_1,\\ldots,I_{q+1})$ which satisfies \\eqref{corr count 6}--\\eqref{corr count 5} simultaneously.\n\n\n\n\nFor each $G_i^j(\\phi,I_r)$ with $r\\in[q]$, we aim to apply Lemma~\\ref{lem: random F decomp} to the family $\\cF:=R_{G_i^j(\\phi,I_r)}(F)$ with $5\\eps,\\delta\/2$ playing the role of $\\eps, \\delta$ respectively. Condition \\ref{F decomp 3} is satisfied due to \\eqref{corr count 4}, and \\ref{F decomp 4} and \\ref{F decomp 5} are satisfied due to \\eqref{corr count 3}, \\eqref{corr count 4} and the fact that $r_{G_i^j(\\phi,I_r)}(F,uw\\cup u'w')\\le h^2n^{f-3}$.\\COMMENT{\n$$(1\\pm 5\\eps)\\frac{|\\cF(v)|}{|\\cF(uw)|}\\stackrel{\\eqref{corr count 4}}{=} \\frac{f|E|}{h|V|},$$\nthus we have \\ref{F decomp 3}.\nNote that $f\\geq 3$ and $| E(G^j_i(\\phi,I_r)) | \\leq (n\/q)^2 $ and $|V(G_i^j)| \\geq n\/q$.\nHence for any $uw\\in E$ we have\n$$|\\cF(uw)|\\stackrel{\\eqref{corr count 3},\\eqref{corr count 4}}{\\geq}\n\\frac{ d^hp_r^h (n\/b)^{f-1}}{ 5\\frac{f| E(G^j_i(\\phi,I_r)) |}{ h|V(G_i^j)|} }\n\\geq \\frac{ d^hp_r^h (n\/b)^{f-1}}{ \\frac{ 5f (n\/q)^2}{ h n\/q} } \\geq  d^h p_r^{h} (n\/b)^{f-2}\/(5f) \\geq \\frac{n^{f-2}}{(\\log{n})^{h+f}} \\geq 10 (5\\epsilon)^{-1} \\log{n},$$\nthus we have \\ref{F decomp 4}. Since $\\cF(uw,u'w')\\le h^2n^{f-3}$, this also implies \\eqref{F decomp 5}.\n}\nDue to \\eqref{eqcolor} and the boundedness of the colouring, for each colour $c\\in I_r$ we have\n$$|E(G_i^j(\\phi,c))|\\le \\frac{f(f-1)(1-\\alpha)\\frac{fdn}{2h} \\pm\\zeta^{1\/2} n}{b^2}\\le \\frac{(1-2\\gamma){f\\choose 2}fdn}{b^2h}\\le \\frac{(1-\\gamma)p_r{f\\choose 2}dn}{bh}\\le \\frac{f|E(G_i^j(\\phi,I_r))|}{h|V(G_i^j)|},$$\nwhere the final inequality follows from the second equality in \\eqref{corr count 2} and \\eqref{corr count 6}. Note that this is the only place where we make full use of the (global) boundedness condition on the colouring.\nThus, for any $v\\in V(G_i^j)$ and $c\\in I_r$, the number of rainbow copies of $F$ in $G_i^j(\\phi,I_r)$ containing  an edge of colour $c$ is at most $$|E(G_i^j(\\phi,c))|\\cdot \\max_{uw\\in E(G_i^j)}\\big\\{r_{G_i^j(\\phi,I_r)}(F,uw)\\big\\}\\overset{\\eqref{corr count 4}}{\\le} \\frac1{1- 5\\eps} r_{G_i^j(\\phi,I_r)}(F,v),$$ and condition \\ref{F decomp 2} is satisfied. Finally, the verification of the codegree assumptions \\ref{F decomp 1} uses \\eqref{corr count 7} and can be done as in the proof of Theorem~\\ref{thm: approx decomp}. We present only the calculation for the codegree of two colours $c,c'\\in[m]$. Recall that due to \\ref{U4} the colouring of $G_i^j$ is locally $\\ell$-bounded with $\\ell:=\\eta^{1\/4}p_r^3n$. Then,  for $c\\neq c'\\in [m]$ and $w\\in V(G_i^j)$, we have\n\\begin{small}\\begin{align*}\n\\hspace{-1cm} |\\{ \\bar F\\in R_{G_i^j(\\phi,I_r)}(F):  \\{c,c'\\} \\subseteq \\phi(E(\\Bar{F}))\\}| &\\overset{\\eqref{corr count 7}}{\\leq}\n\\sum_{uv\\in E(G_i^j(\\phi,c))} \\big(d_{G_i^j(\\phi,c')}(u)+d_{G_i^j(\\phi,c')}(v)\\big) 2f! p_r^{h-3}\\Big(\\frac nq\\Big)^{f-3} \\nonumber\\\\\n+ \\sum_{uv \\in E(G_i^j(\\phi,c'))} \\sum_{ u'v' \\in E(G_i^j(\\phi,c)-\\{u,v\\})} 2f! \\Big(\\frac nq\\Big)^{f-4}\n&\\overset{\\eqref{eqcolor}}{\\leq} 2f^3\\frac n{b^2} \\cdot 2\\ell \\cdot 2f! p_r^{h-3}\\Big(\\frac nq\\Big)^{f-3} + 2(fn)^2\\cdot f! \\Big(\\frac nq\\Big)^{f-4} \\nonumber\\\\\n&\\leq \\eta^{1\/5} p_r^h \\Big(\\frac nb\\Big)^{f-1}  \\overset{\\eqref{corr count 3}}{<}\n\\eta^{1\/6}|R_{G_i^j(\\phi,I_r)}(F,w)| .\n\\end{align*}\n\\end{small}\nThe other calculations can be done similarly\n\\COMMENT{Note that we have $n\/q$  instead of $n$ in Theorem~\\ref{thm: approx decomp} (more precisely, we have $f$ parts of size $n\/b$ each). The coloring is locally $\\ell$-bounded, where $\\ell \\le \\eta^{1\/4} p_r^3 n$, $\\ell\\le \\eta^{1\/4}p_r^2n\/b$. (see \\ref{U4}).\\\\\nFor each $u \\neq v \\in V$, there are at most $f! n^{f-2}$ copies of $F$ containing both $u$ and $v$, so we have\n\\begin{align*}\n|R_{G_i^j}(F,u)\\cap R_{G_i^j}(F,v)| \\leq f! n^{f-2}\\overset{\\eqref{corr count 3}}{<}\\eta |R_{G_i^j(\\phi,I_r)}(F,w)|.\n\\end{align*}\nNote that by \\eqref{eqcolor} and \\eqref{colorbound} we have $e(G_i^j(\\phi,c))\\le 2f^3n\/b^2$ for any $c\\in[m]$. Using the conclusions of Lemma~\\ref{lem: colour partition} for any $uw\\in E(G_i^j))$ and with $\\cF:=\\{F'-uw: F'\\in R_{G_i^j}(F,uw)\\}$ playing the role of $\\cF$, we have $R_{G_i^j(\\phi, I_r)}(F,uw)=(1\\pm \\eps)p_r^{h-1}R_{G_i^j}(F,uw)\\le 2f!p^{h-1}_r (n\/q)^{f-2},$ where the first equality holds for all $i\\in [g]$, $j\\in [q]$ and $uv\\in E(G_i^j)$. Note we have obtained it for \\eqref{corr count 4}, but it is not explicitly stated there (so formally, we would need to at this as\n \\eqref{corr count 4}(b)).\nThus, one can obtain that for $c\\in [m]$ and $v\\in V(G_i^j)$ one has \\begin{align*}\n\\hspace{-1cm} &|\\{ \\Bar{F}\\in R_{G_i^j(\\phi, I_r)}(F):  v\\in \\Bar{F},c \\in \\phi(E(\\Bar{F}))\\}| \\leq\n\\sum_{u\\in N_{G_i^j(\\phi,c)}(v)} 2f! p_r^{h-1}(n\/q)^{f-2} \\nonumber\\\\\n&+ \\sum_{uw \\in E(G_i^j(\\phi,c))} f! (n\/q)^{f-3}\n\\leq 2\\ell f! p_r^{h-1}(n\/q)^{f-2} + fn\\cdot f! (n\/q)^{f-3} \\leq \\eta^{1\/5} p_r^h n^{f-1}\/q^{f-1}.\n\\end{align*}\n Finally, note that $\\eta^{1\/5} p_r^h n^{f-1}\/q^{f-1}\\le \\eta^{1\/6}|R_{G_i^j(\\phi,I_r)}(F,w)|$ due to \\eqref{corr count 3}.}\n\n\nThus, we conclude that, for all $i\\in [g],$ $j,r\\in[q]$, there is a randomized algorithm  which returns a $\\delta$-decomposition of $G_i^j(\\phi,I_r)$ into rainbow $\\delta\/2$-spanning $F$-factors, such that each $v\\in V(G_i^j)$ belongs to each factor with probability at least $1-\\delta\/2$. By \\ref{U2} and \\eqref{corr count 6}  we may assume that the number of $\\delta\/2$-spanning $F$-factors in the $\\delta$-decomposition of $G_i^j(\\phi,I_r)$ is the same for all $i,j,r$.\nWe denote this number by $n_\\delta$.\nFor each $i\\in[g],$ $ j,r\\in[q]$, delete a randomly chosen collection of $\\frac{\\delta|V(G_i^j)|}{3f}$ copies of $F$ from each of the $\\delta\/2$-spanning $F$-factors\nof $G_i^j(\\phi,I_r)$.\nThen the proportion of vertices of $V(G_i^j)$ covered by each factor is at least\n $1-\\delta$ and at most $1-\\delta\/3$, and  each $v\\in V(G_i^j)$ belongs to each factor with probability at least $1-\\delta$.\n Moreover, the factors clearly form a $2\\delta$-decomposition of $G_i^j(\\phi,I_r)$.\n For $k'\\in [n_{\\delta}]$, $j\\in [q]$ and $i\\in[g]$, let $\\cD_i^j(k', I_r)$ denote the resulting $k'$-th $\\delta$-spanning $F$-factor in this $2\\delta$-decomposition of $G_i^j(\\phi,I_r)$. Note that the total number of edges in the $\\cD_i^j(k',I_r)$ over all $i\\in[g], j,r\\in [q]$ and $k'\\in [n_{\\delta}]$ is at least $$\\sum_{i\\in [g]}\\sum_{j,r\\in[q]} n_{\\delta}\\cdot (1-\\delta)\\frac hf|V(G_i^j)|=(1-\\delta)\\frac hfn \\sum_{i\\in [g]}\\sum_{r\\in[q]} n_{\\delta}= (1-\\delta)\\frac hfn \\cdot gqn_\\delta,$$\n but, on the other hand, is at most ${n\\choose 2}$, and therefore\n \\begin{equation}\\label{ndelta} gqn_{\\delta} \\leq \\frac fh n.\\end{equation}\n Summarizing, these almost-spanning $F$-factors satisfy the following properties.\n\\begin{itemize}\n  \\item[\\textbf{a)}] For all $i\\in[g],\\ j,r\\in[q]$, $k'\\in [n_{\\delta}]$ and $v\\in V(G_i^j)$, we have $v\\in \\cD_i^j(k', I_r)$ with probability at least $1-\\delta$.\n  \\item[\\textbf{b)}] For all $i\\in [g]$ and $j_1,j_2,r_1,r_2\\in[q]$ with $(j_1,r_1)\\ne (j_2,r_2),$ the random variables $\\cD_i^{j_1}(k', I_{r_1})$ and $\\cD_i^{j_2}(k', I_{r_2})$ are independent.\n\\end{itemize}\n For all $i \\in [g]$, $r\\in[q]$, $k'\\in[n_{\\delta}]$ put\n \\begin{equation}\\label{eqfullspan}\\cD_i(k'+(r-1)n_{\\delta}):=\\bigcup_{j=1}^q \\cD_i^j(k', I_{r+j}),\\end{equation} where the index $r+j$ is modulo $q$. It is easy to see that for each $i\\in[g]$ and $k\\in[qn_{\\delta}]$ the family $\\cD_i(k)$ is a rainbow $\\delta$-spanning $F$-factor in $G$, and that $\\bigcup_{k=1}^{qn_\\delta} \\cD_i(k) = \\bigcup_{k'=1}^{n_{\\delta}}\\bigcup_{j,r=1}^q \\cD_i^j(k',I_r)$.\nMoreover, the $\\cD_i(k)$ are pairwise edge-disjoint.\nDenote by $V'(\\cD_i(k))$ the set of vertices not covered by $\\cD_i(k)$. We have\n\\begin{equation}\\label{bigremainder} \\frac {\\delta}3 n\\le |V'(\\cD_i(k))|\\le \\delta n.\n\\end{equation}\nWe claim that the following properties hold with high probability.\n\\begin{enumerate}\n\\item[\\textbf{A}]\\label{random alg 1} For any $v\\in V(G)$ we have $v\\in V'(\\cD_i(k))$ for at most $2f\\delta n$ choices of $(k,i)\\in[qn_{\\delta}] \\times [g]$.\n\\item[\\textbf{B}]\\label{random alg 2} For any $c\\in I_{q+1}$, $i\\in [g]$ and $k\\in [qn_{\\delta}]$ the number of edges of colour $c$ incident to $V'(\\cD_i(k))$ is at most $3f\\delta n$.\n\\end{enumerate}\n By Lemma~\\ref{Chernoff} and properties \\textbf{a)}, \\textbf{b)}, for any $i\\in[g]$, $j\\in [q]$, $v\\in V(G_i^j)$ and $k'\\in [n_{\\delta}]$, the probability that there are at least $2\\delta q$ indices  $r\\in[q]$ such that $v\\notin \\cD_i^j(k',I_r)$ is at most $2e^{-\\delta q\/3}\\le n^{-4}$. Now a union bound shows that with probability at least $1-o(1)$ for all $i\\in[g]$, $j\\in [q]$, $v\\in V(G_i^j)$ and $k'\\in [n_{\\delta}]$ there are at most $2\\delta q$ indices $r\\in [q]$ such that $v\\notin \\cD_i^j(k',I_r)$.\n Thus, using \\eqref{ndelta}, with probability $1-o(1)$ every vertex $v\\in V(G)$ belongs to all but at most $2\\delta q g n_{\\delta}\\le 2f\\delta n$ of the $\\delta$-spanning factors $\\cD_i(k)$. Consequently, \\textbf{A} holds with high probability.\n\nFix $c\\in I_{q+1}$, $k'\\in[n_{\\delta}],$ $i\\in [g]$ and $r\\in [q]$ and put $k:=k'+(r-1)n_{\\delta}$. Define the martingale $X_0,\\ldots, X_q$, where $X_{j'}$ is equal to the expected number of edges of colour $c$ incident to $V'(\\cD_i(k))$, given the choices of the almost-spanning factors $\\cD_i^1(k',I_{r+1})$, \\ldots, $\\cD_i^{j'}(k',I_{r+j'})$, with indices taken modulo $q$  (cf. \\eqref{eqfullspan}).  We have $X_{0}<2f\\delta n$ due to \\textbf{a)} and \\eqref{colorbound}.\nMoreover, $|X_{j'}-X_{j'-1}|\\le 4f^2n\/b$ due to \\eqref{colorbound}, \\eqref{eqcolor} and \\textbf{b)}.\\COMMENT{Properties \\eqref{colorbound} and \\eqref{eqcolor} imply that the number of edges of colour $c$ incident to $V(G_i^j)$ is at most $fn\\cdot 4\/b^2\\cdot bf=4f^2n\/b$. Property \\textbf{b)} is needed because we do not want to affect the future decisions (and thus potentially change the value of $X_i$ because of that).} Thus, using Theorem~\\ref{Azuma}, we obtain $\\mathbb P\\big[X_q\\ge 3f\\delta n\\big]\\le n^{-5}$,\\COMMENT{\nThe probability is at most the probability that $P\\big[|X_q-X_0|\\ge f\\delta n]\\le  e^{-\\frac{f^2\\delta^2n^2}{2\\sum_{j'=1}^q (4f^2n\/b)^2}}= e^{-c_f\\delta^2b}\\le e^{-\\eta^{-1\/(4a)}\\log n}<n^{-5}$, where $c_f$ depends only on $f$.\n}\n and with probability at least $1-n^{-1}$ none of these events for different $i\\in [g],k'\\in [n_\\delta], r\\in[q],c\\in I_{q+1}$ occurs.\n Hence we can choose the $\\cD_i^j(k',I_r)$ such that \\textbf A and  \\textbf{B} hold.\n\n\n\n\n\n\n\n\n\n\n\nThe remaining part of the proof is concerned with turning $\\cD_i(k)$ for all $i\\in[g]$, $k\\in [qn_{\\delta}]$ into a {\\it spanning} $F$-factor using  the rainbow blow-up lemma (Lemma~\\ref{blowup}). We cannot apply Lemma~\\ref{blowup} to $V'(\\cD_i(k))$ directly, so we add some random vertices to it as described below. Fix $i\\in [g]$ and $k\\in [qn_{\\delta}]$. Define a random subcollection $\\mathcal C_{i,k}$ of $\\{G_i^j: j\\in[q]\\}$ as follows. \\begin{itemize}\n\\item[\\textbf{c)}] Include each $Q\\in \\{G_i^j: j\\in[q]\\}$ into $\\mathcal C_{i,k}$ independently at random  with probability $\\delta_1$.\n\\end{itemize}\nPut\n$$V_1(i,k):=\\bigcup_{Q\\in \\mathcal C_{i,k}} V(Q), \\ \\ \\ \\ \\ \\ \\  V_2(i,k):=V'(\\cD_i(k))\\setminus V_1(i,k).$$\nRecall that each $Q\\in \\mathcal C_{i,k}$ is $f$-partite and for each $j'\\in [f]$ let $V_1^{j'}(i,k)$ be the union over all $Q\\in \\mathcal C_{i,k}$ of the $j'$-th vertex class of $Q$. In particular, $\\bigcup_{j'=1}^f V_1^{j'}(i,k) = V_1(i,k).$ For every $k\\in [qn_{\\delta}]$ and $i\\in[g]$, put\n$$W'(i,k):=G[V_1^1(i,k),\\ldots, V_1^{f}(i,k),V_2(i,k)](\\phi, I_{q+1}).$$\n\n\n\n\nConsider an arbitrary $F$-factor $H_{i,k}$ on $[N_{i,k}]$, where $N_{i,k}:=|V_1(i,k)|+|V_2(i,k)|$. Note that $N_{i,k}$ is divisible by $f$ since $n$ is divisible by $f$ and $\\cD_i(k)\\setminus V_1(i,k)$ is an $F$-factor on $V(G)\\setminus (V_1(i,k)\\cup V_2(i,k))$. Recall that in the case when $F$ was an edge, we had to replace it with two disjoint edges. If $n$ (and thus also $N_{i,k}$)  is not divisible by $4$, let $H_{i,k}$ be the union of an $F$-factor on $[N_{i,k}-2]$ and the edge $\\{N_{i,k}-1,N_{i,k}\\}$. Split $V(H_{i,k})$ arbitrarily into $f+1$ independent sets $S_0,\\ldots, S_{f}$, where $|S_{j'}|=|V_1^{j'}(i,k)|$ for each $j'\\in [f]$, and where $|S_0|=|V_2(i,k)|$. Moreover, we require that each copy of $F$ in $H_{i,k}$ intersects each of $S_0,\\ldots, S_f$ in at most one vertex. Using \\ref{U1} and \\eqref{bigremainder}, it is easy to see that such a partition always exists provided $N_{i,k}\\ge \\delta_1 n\/2,$ say (which will be satisfied by  $\\mathbf{C_i}$ below).\\COMMENT{\nWe use the fact that the number of vertices in each $U_j$ is very tightly concentrated, and thus the difference between different $V_1^{i'}$ (roughly $\\zeta n$) is much smaller than $|V_2|$ (roughly $\\delta n$, see \\eqref{bigremainder}). More precisely, assume that $V_1^1$ is the largest out of $V_1^{i'}$, and put $x_{i'}:=|V_1^1|-|V_1^{i'}|$. Then, due to \\ref{U1} and the definition of $V_1^{i'}$, we have $x_{i'}\\le 2\\zeta n$. Thus, we have $\\sum_{i'=1}^f x_{i'}\\le |V_2|\\sim \\delta n$. For each $i'\\in [f]$ take $x_{i'}$ copies of $F$, which are not yet assigned to the parts $S_0,\\ldots, S_f$, and assign each vertex to a distinct part, excluding part $i'$. Once we are done for all $i'$, we see that the number $N$ of vertices we still need to assign to the parts is the same for all parts $S_1,\\ldots, S_f$, moreover, the remaining number $N_0$ of vertices to be assigned to $S_0$ is divisible by $f$. Note $N_0\\le N$. Now for each $i'\\in [f]$ take $N_0\/f$ copies of $F$ and assign one vertex of each of these copies to each part apart from $S_{i'}$. After that $S_0=\\emptyset$ and the number of $V_i$ we still need to assign to $S_1,\\ldots, S_f$ is the same.\n}\n\n\nNote that both $V_1(i,k)$ and $N_{i,k}$ are  still  random variables at this stage. For all pairs $(i,k)$ in lexicographical order, where $i\\in [g]$ and $k\\in [qn_{\\delta}]$, we proceed iteratively as follows. Fix $i\\in [g]$. Assume that we have already fixed a choice of all the $V_1(i^*,k^*)$ for $i^*<i$ and $k^*\\in [qn_{\\delta}]$ and that we have constructed edge-disjoint embeddings $\\psi_{i^*,k^*}: H_{i^*,k^*}\\to W'(i^*,k^*)$ for all $i^*<i$, $k^*\\in[qn_{\\delta}]$, which satisfy $\\mathbf {C_1},\\ldots, \\mathbf{C_{i-1}},\\mathbf{D_1},\\ldots,\\mathbf{D_{i-1}},\\mathbf{E_1},\\ldots, \\mathbf{E_{i-1}}$ below. Define the graph\n$$\nK_{i}:=\\bigcup_{i^*=1}^{i-1}\\bigcup_{k^*=1}^{qn_{\\delta}}\\psi_{i^*,k^*}(H_{i^*,k^*}).\n$$\n Next, we choose $V_1(i,k)$ for all $k\\in [qn_{\\delta}]$  simultaneously. We claim that there exists a choice of these $V_1(i,k)$ such that the following hold.\n\\begin{itemize}\n  \\item[$\\mathbf {C_i}$] For each $k\\in [qn_{\\delta}]$ we have $|V_1(i,k)|=(1\\pm \\delta_1)\\delta_1n.$\n  \\item[$\\mathbf {D_i}$] Each $v\\in V(G)$ belongs to at most $2\\delta_1 qn_{\\delta}$ sets among $V_1(i,1),\\ldots, V_1(i,qn_{\\delta})$.\n\\item[$\\mathbf {E_i}$] For all $v\\in V(G)$ and $k\\in [qn_{\\delta}]$ we have $d_{K_{i},V_1(i,k)}(v)\\le \\delta_1^{3\/2}n$.\n\\end{itemize}\nTo prove the claim, note that $\\EXP[|V_1(i,k)|]=\\delta_1n$. Recall from \\ref{U1} that for all $i_1,j_1,i_2,j_2$ we have  $|V(G_{i_1}^{j_1})|=(1\\pm 2\\zeta)|V(G_{i_2}^{j_2})|.$\nUsing Lemma~\\ref{Chernoff} and \\textbf{c)}, it follows that $\\mathbf{C_i}$ is satisfied with probability at least $1-e^{-\\delta_1^3q\/3}\\ge 1-n^{-2}$ for fixed $k$.\\COMMENT{\nit is at least $1-2e^{-\\frac{(0.99\\delta_1^2 q)^2}{2(\\delta_1 q+(0.99\\delta_1^2 q)\/3)}}$, where we replaced $1$ by $0.99$ to compensate for the fact that $G_i^j$ are not of exactly the same size.\n} Taking a union bound over all $k\\in [qn_{\\delta}]$, we conclude that $\\mathbf{C_i}$ is satisfied with probability $1-o(1)$.\n\nNext, using \\textbf{c)} and Lemma~\\ref{Chernoff}, for any $j\\in [q]$ the probability that $G_i^j$ belongs to $\\mathcal C_{i,k}$ for at least $2\\delta_1 qn_{\\delta}$ different values of $k$ is at most $e^{-\\delta_1 q n_{\\delta}\/3}$. Taking a union bound over all $j$, we conclude that  $\\mathbf{D_i}$ holds with probability $1-o(1)$.\n\n\nFinally, using \\textbf{A}, as well as $\\mathbf{D_{i^*}}$ for $i^*<i$ and the fact that  each $H_{i^*,k^*}$ has maximum degree at most $f$,  we conclude that  $d_{K_i}(v)\\le 2f^2\\delta n+f(i-1)\\cdot 2\\delta_1q n_{\\delta}\\le 4f^2\\delta_1 n$. Fix $v\\in V(G)$ and $k\\in [qn_{\\delta}]$. Similarly to the proof of \\textbf{B}, define the martingale $X_0,\\ldots, X_q$, where $X_t:=\\Exp\\big[d_{K_{i},V_1(i,k)}(v)\\mid \\mathcal C_{i,k}\\cap \\{G_i^1,\\ldots, G_i^{t}\\}\\big]$. We have $X_0\\le 4f^2\\delta_1^2 n$ and $|X_t-X_{t-1}|\\le \\vartheta_t$, where $\\vartheta_t\\le |V(G_i^t)|\\le 2n\/q$. Thus, $\\sum_{t=1}^q \\vartheta_t^2\\le  \\frac{2n^2}q$. Applying Theorem~\\ref{Azuma}, we obtain that $\\Pro[X_q\\ge \\delta_1^{3\/2} n]\\le n^{-4}$ and,\\COMMENT{The probability is at most $2e^{\\frac{-\\delta_1^3n^2}{4n^2\/q}}\\le 2e^{-\\delta_1^3q\/4}\\le n^{-4}.$} taking a union bound over all choices of $v\\in V(G)$ and $k\\in [qn_{\\delta}]$, we conclude that $\\mathbf{E_i}$ holds with probability $1-o(1)$.\nFix choices of $V_1(i,1),\\ldots, V_1(i,qn_{\\delta})$ that satisfy properties $\\mathbf{C_i}$, $\\mathbf{D_i}$, $\\mathbf{E_i}$ simultaneously.\n\nWe remark that $\\mathbf{C_i}$ together with \\ref{U1} and \\eqref{bigremainder} imply that\n \\begin{align}\\label{eqsize12} |\\{j\\in[q]:G_i^j\\in \\mathcal C_{i,k}\\}|&=(1\\pm 2\\delta_1)\\delta_1q, \\ \\ \\ \\ \\ \\ |V_1^{j'}(i,j)|=(1\\pm 2\\delta_1)\\delta_1 n\/f\\ \\ \\ \\ \\text{and} \\\\\n \\label{eqsize11}\n  \\ \\ \\ \\ |V_2(i,k)|&\\le \\delta_1|V_1^{j'}(i,k)|\\ \\ \\ \\ \\ \\ \\ \\  \\text{ for each } j'\\in[f].\\end{align}\n For each $k\\in[qn_{\\delta}]$ in turn, we now intend to apply Lemma~\\ref{blowup} using $H_{i,k}$ with partition $S_0,\\ldots, S_f$  and an arbitrary bijection $\\psi'_{i,k}: S_0\\to V_2(i,k)$ and with $\\delta_1n\/f,f,\\gamma d,\\delta_2,\\gamma$ playing the roles of $n,r,d,\\delta_2,\\gamma$ to\n$$W(i,k):=W'(i,k)-K_{i}-\\bigcup_{k^*=1}^{k-1}\\psi_{i,k^*}(H_{i,k^*})$$\nwith partition $V_2(i,k),V_1^1(i,k),\\dots, V_1^f(i,k)$. Provided that such an application is possible, we can extend $\\psi'_{i,k}$ to $\\psi_{i,k}$ and obtain a rainbow $F$-factor $\\psi_{i,k}(H_{i,k})$ in $W(i,k)$ for each $i,k$. (Indeed, the graph $\\psi_{i,k}(H_{i,k})\\cup (\\cD_i(k)\\setminus V_1(i,k))$ forms a rainbow $F$-factor in $G'$.) Moreover, $\\cH:=\\bigcup_{i,k}\\big(\\psi_{i,k}(H_{i,k})\\cup (\\cD_i(k)\\setminus V_1(i,k))\\big)$ gives us an $\\alpha$-decomposition of $G'$, as required. Indeed, using \\ref{U2}, \\eqref{corr count 5}, \\eqref{eqfullspan} and the fact that $\\{\\cD_i^j(k',I_r):k'\\in[n_{\\delta}]\\}$ forms a $2\\delta$-decomposition of $G_i^j(\\phi,I_r)$, it is easy to see that  already $\\bigcup_{i,k}\\cD_i(k)$ covers all but an $\\gamma^{1\/2}$-fraction of the edges of $G'$,\\COMMENT{\nBy definition, $\\bigcup_{i,k}\\cD_i(k)$ covers all but all but a $2\\delta$-proportion of the edges in $G[U_1,\\ldots, U_b](\\phi, [m]\\setminus I_{q+1})$ (and in particular it has at least $\\frac d3 n^2$ edges by \\ref{U2}). On the other hand, $|e(G')-e(G)|\\le 2\\zeta n^2$ and $e(G)-e(G[U_1,\\ldots, U_b])\\le 2\\zeta n^2$. Finally, \\eqref{corr count 5} implies that  $e(G[U_1,\\ldots, U_b])-e(G[U_1,\\ldots, U_b](\\phi,[m]\\setminus I_{q+1}))\\le 2\\gamma dn^2$. Since $\\zeta\\ll \\gamma\\ll d$, we conclude that $G[U_1,\\ldots, U_q](\\phi,[m]\\setminus I_{q+1})$ contains all but a $\\gamma^{2\/3}$-proportion of edges of $G'$, and, combining, we get the claim.\n}\nand $\\cH$ contains at least as many edges (as it consists of spanning rather than almost-spanning factors). Thus, to complete the proof, we only need to\nverify that Lemma~\\ref{blowup} is applicable in each iteration step. Indeed, we have the following.\n\\begin{itemize}\n  \\item Property \\textbf B,  combined with \\eqref{colorbound}, \\eqref{eqcolor} and the first part of \\eqref{eqsize12} imply that the colouring $\\phi$ of $W(i,k)$ is $3f\\delta n + \\frac{3fn}{b^2}\\cdot (2\\delta_1q)^2{f\\choose 2}\\le \\delta_2(\\delta_1n\/f)$-bounded.\n  \\item\\ref{lem blowup 1} is implied by the definitions of $H_{i,k}$ and $S_0$, with $\\Delta=f$.\n\\item \\ref{lem blowup 2} is  implied by the definition of $\\psi'_{i,k}$ and  \\eqref{eqsize11}.\n\\item \\ref{lem blowup 3} is implied by  \\eqref{eqsize12}, together with the definition of $H_{i,k}$.\n \n  \\item By \\eqref{corr count 5}, for $j_1\\ne j_2\\in[f]$ the graph $W'(i,k)\\big[V_1^{j_1}(i,k),V_1^{j_2}(i,k)\\big]$ is $(\\delta_1,\\gamma d)$-superregular and for each $v\\in V_2(i,k)$, $j'\\in[f]$ we have $d_{W'(i,k),V_1^{j'}(i,k)}(v)=\\gamma d|V_1^{j'}(i,k)|\\pm \\delta_1|V_1^{j'}(i,k)|.$ Let\n      $$K_i^k:=K_i \\cup \\bigcup_{k^*=1}^{k-1}\\psi_{i,k^*}(H_{i,k^*}).$$\n  Then, due to $\\mathbf{E_i}$, for each vertex $v\\in V(G)$ we have $$d_{K_{i}^k,V_1(i,k)}(v)\\le d_{K_{i},V_1(i,k)}(v)+fqn_{\\delta}\\le \\delta_1^{1\/3}\\cdot(\\delta_1n\/f).$$\n  (Note that we bounded the contribution of $\\bigcup_{k^*=1}^{k-1}\\psi_{i,k^*}(H_{i,k^*})$ by $fqn_{\\delta}\\le f^2n\/g\\le \\delta n$, using \\eqref{ndelta}.)\n   Using Proposition~\\ref{prop: edge deletion regular}, we conclude that for all $j_1,j_2,j'\\in[f]$ and $v\\in V_2(i,k)$ $$W(i,k)[V_1^{j_1}(i,k),V_1^{j_2}(i,k)]\\text{ is }(\\delta_2,\\gamma d)\\text{-superregular and }\\ d_{W(i,k),V_1^{j'}(i,k)}(v)\\ge \\frac 12\\gamma d|V_1^{j'}(i,k)|.$$\n   Thus,~\\ref{lem blowup 4} and~\\ref{lem blowup 5} are satisfied.\n \\end{itemize}\n This concludes the proof of Theorem~\\ref{thm: perfect decomp}.\n\n\n\\subsection{Proof of Theorem~\\ref{thm: spanning cycle}. Almost-decomposition into Hamilton cycles}\nThe proof of this theorem is very similar to that of Theorem~\\ref{thm: perfect decomp}. In particular, we use the same notation as in the proof of Theorem~\\ref{thm: perfect decomp}. We will let a cycle $C_s$ with sufficiently large $s$ play the role of $F$. (We remark that $a(C_s)=2$.)  We then merge each almost-spanning $C_s$-factor into a single Hamilton cycle. This introduces a final ``gluing'' step, and, in particular, changes the graphs $H_{i,k}$ and embeddings $\\psi'_{i,k}$ we use. This part of the proof resembles the final part of the proof of Theorem~\\ref{thm: near spanning cycle}. The main difference to Theorem~\\ref{thm: near spanning cycle} is that we have to include all the vertices into the  cycle this time.\n\nLet us make this precise.  We use the following hierarchy of constants: \\begin{small}\\begin{equation\n0<1\/n_0\\ll\\eta\\ll \\zeta\\ll \\zeta_1\\ll\\epsilon\\ll\\eps_1\\ll \\delta\\ll \\delta_1\\ll \\delta_2\\ll 1\/s\\ll\\delta_3\\ll\\delta_4\\ll \\gamma\\ll \\beta\\ll\\alpha, d_0.\\end{equation}\\end{small}\nNote that the position of $f,h$ in the hierarchy in the proof of  Theorem~\\ref{thm: perfect decomp} is consistent with $f=h=s$ and \\eqref{eqhier}. We additionally assume that $s$ is even.\n\n\nWe proceed until the stage just before properties \\textbf{A} and \\textbf{B}. In particular, for all $i\\in [g]$, $k\\in[qn_{\\delta}]$ we define a $\\delta$-spanning $C_s$-factor $\\cD_i(k)$ in $G$.  We write $\\cD_i(k)=\\bigcup_{j=1}^{n_2}C_i^j(k),$ where $C_i^j(k)$ are the $s$-cycles forming $\\cD_i(k)$. Moreover, by (randomly) disregarding some cycles if necessary, we assume that the number $n_2$ is the same for all $i,k$.\nFor every $k\\in [qn_{\\delta}]$ we define the following sets of vertices:\n\\begin{align*}\n V'(\\cD_i(k)):=& V(G)\\setminus V(\\cD_i(k))\\ \\ \\ \\ \\ \\ \\text{and}\\\\\nV''(\\cD_i(k)):=&\\bigcup_{j=1}^{n_2}\\{x_i^j(k),y_i^j(k)\\},\\ \\  \\ \\text{ where }x_i^j(k)y_i^j(k)\\text{ is an edge, randomly chosen from }C_i^j(k).\\end{align*}\n\nWe modify the properties \\textbf{A} and \\textbf{B} accordingly. We claim that the following hold with high probability.\n\\begin{enumerate}\n\\item[\\textbf{A}]\\label{random alg 1} For any $v\\in V(G)$ we have $v\\in V'(\\cD_i(k))\\cup V''(\\cD_i(k))$ for at most $2s\\delta n+4n\/s$ choices of $(k,i) \\in[qn_{\\delta}] \\times [g]$.\n\\item[\\textbf{B}]\\label{random alg 2} For all $c\\in I_{q+1}$, $i\\in [g]$ and $k\\in [qn_{\\delta}]$ the number of edges of colour $c$ incident to $V'(\\cD_i(k))\\cup V''(\\cD_i(k))$ is at most $3s\\delta n+4n\/s$.\n\\end{enumerate}\nIn view of the proof of Theorem~\\ref{thm: perfect decomp}, we only have to verify the parts of the properties \\textbf A and \\textbf B involving $V''(\\cD_i(k))$.\n\nFix $v\\in V(G)$. To verify the second part of \\textbf{A}, note that for any $i\\in[g]$ the number $\\rho_v$ of sets among $V''(\\cD_i(1)),\\ldots, V''(\\cD_i(qn_{\\delta}))$ to which $v$ belongs is a random variable, which is a sum of $qn_{\\delta}$ independent binary random variables with probability of success at most $2\/s$. Since\n$qn_{\\delta}\\cdot 2\/s\\le 2n\/gs$ due to \\eqref{ndelta}, a standard application of  Lemma~\\ref{Chernoff} (together with a union bound over $i \\in [g]$) implies that the second part of \\textbf{A} holds for all $v\\in V(G)$ with probability $1-o(1)$.\n\nIn order to ensure that for all $c\\in I_{q+1}$, $i\\in [g]$, $k\\in[qn_{\\delta}]$ the number of edges of colour $c$ incident to $V''(\\cD_i(k))$ is at most $4n\/s$, we define a martingale $X_0,\\ldots, X_{n_2}$, where $X_{j}$ is equal to the expected number of edges of color $c\\in I_{q+1}$ incident to $V''(\\cD_i(k))$ given the choices of random edges in $C_i^1(k),\\ldots, C_i^{j}(k)$. By \\eqref{colorbound} we have $X_0\\le 2n\/s$ and, since the coloring $\\phi$ is locally $\\eta n\/\\log^4 n$-bounded,  $|X_j-X_{j-1}|\\le 2\\eta \\frac n{\\log^4 n}$. Putting $c_i^j(k)$ to be the number of edges of colour $c$ incident to $C_i^j(k)$, we also get that $\\sum_{j=1}^{n_2}|X_j-X_{j-1}|\\le \\sum_{j=1}^{n_2} c_i^j(k)\\le 2n$. Thus, Theorem~\\ref{Azuma} implies that $\\mathbb{P}\\big[|X_{n_2}-X_0|\\ge 2n\/s]\\le  n^{-5}.$\\COMMENT{$\\le 2e^{-\\frac{(2n\/s)^2}{4\\eta \\frac n{\\log^4n}\\cdot 2n}}\\le 2e^{-\\log^4n} \\le$}\nThus, \\textbf B holds with high probability.\nFix a choice of the $\\cD_i(k)$ that satisfies \\textbf{A} and \\textbf{B} simultaneously.\n\n\nFix $i\\in [g]$ and $k\\in [qn_{\\delta}]$. Define a random subcollection $\\mathcal C_{i,k}$ of $\\{G_i^j: j\\in[q]\\}$ as follows. \\begin{itemize}\n\\item[\\textbf{c)}] Include each $Q\\in \\{G_i^j: j\\in[q]\\}$ into $\\mathcal C_{i,k}$ independently  with probability $\\delta_3$.\n\\end{itemize}\n(Note that the probability in this case is not the same as in the proof of Theorem~\\ref{thm: perfect decomp}.)\nPut\n$$V_1(i,k):=\\bigcup_{Q\\in \\mathcal C_{i,k}} V(Q), \\ \\ \\ \\ \\ \\ \\  V_2(i,k):=V'(\\cD_i(k))\\setminus V_1(i,k) \\ \\  \\ \\text{ and } \\ \\ \\ V_3(i,k):=V''(\\cD_i(k))\\setminus V_1(i,k).$$\n\nRecall that each $Q\\in \\mathcal C_{i,k}$, and thus also $V_1(i,k)$, has $s$ parts and that $s$ is even. Let $V_1^1(i,k)$ be the union of all parts with odd indices and let $V_1^2(i,k):=V_1(i,k)\\setminus V_1^1(i,k)$. Put $n_2(i,k):=|V_3(i,k)|\/2.$\n\nLet $W'(i,k)$ be the following graph:\n$$W'(i,k):=G[V_1^1(i,k), V_1^2(i,k),V_2(i,k),V_3(i,k)](\\phi, I_{q+1}).$$\n\n\nNow we are in a position to define the graph $H_{i,k}$ and the embedding function $\\psi'_{i,k}$. Let $H_{i,k}:=\\bigcup_{j=1}^{n_2(i,k)-1} P^j\\cup P_{i,k}$, where $P^j=w_0^jw_1^j w_2^jw_{3}^j$ is a path of length $3$, and $P_{i,k}=w_0^{n_2(i,k)}\\ldots w_{3}^{n_2(i,k)}$ is a path on $|V(W'(i,k))|-4(n_2(i,k)-1)$ vertices. We also let  $t:=|V_2(i,k)|$ and choose vertices $z_1,\\ldots, z_t$ on $P_{i,k}$, such that the vertices $z_1,\\ldots, z_t,w_0^{n_2(i,k)}, w_{3}^{n_2(i,k)}$ have pairwise distance at least four along the path\nand for all $i' \\in [t-1]$ the distance between $z_i$ and $z_{i+1}$ is precisely four. Clearly, this condition is possible to fulfill since $V_1(i,k)$ is much larger than $V_2(i,k)\\cup V_3(i,k)$ (note that $V_1(i,k)$ has size roughly $\\delta_3n$, while $V_2(i,k)$ and $V_3(i,k)$ have sizes at most $\\delta n$ and $2n\/s$, respectively).  Note that $|V(H_{i,k})|=|V(W'(i,k))|$.\n\n Take a partition of $H_{i,k}$ into three parts $\\cX=\\{X_0,X_1,X_2\\}$, which  satisfies the following: $X_0=\\bigcup_{j=1}^{n_2}\\{w_0^j,w_{3}^j\\}\\cup \\{z_1,\\ldots, z_t\\}$; for $j\\in [2]$ the set $X_j$ is independent in $H_{i,k}$ and has size $|V_1^j(i,k)|$. The final property is easy to satisfy since by \\ref{U1} the sizes of $V_1^1(i,k)$ and $V_1^2(i,k)$ differ by at most $2\\zeta n$, which is much smaller than $t$ due to \\eqref{bigremainder} and thus we have sufficient flexibility in assigning the vertices of $P_{i,k}$ to $X_1$ and $X_2$ since $z_1,\\ldots, z_t$ are assigned to $X_0$. Note that $|X_0|=|V_2(i,k)\\cup V_3(i,k)|$. Moreover, by relabeling if necessary, we may assume that $\\mathcal D_{i}(k) = \\bigcup_{j=1}^{n_2(i,k)} C_i^j(k).$\nWe aim to apply Lemma~\\ref{blowup} with $\\psi'_{i,k}$ defined as follows: $\\psi'_{i,k}(w_0^j)=x_i^j(k)$ and $\\psi'_{i,k}(w_{3}^j)=y_i^{j+1}(k)$, with indices taken modulo $n_2(i,k)$, and $\\psi'_{i,k}(z_j)=v_j$, where $V_2(i,k)=:\\{v_1,\\ldots, v_t\\}$.\n\n\nWe slightly modify the properties $\\mathbf{C_i}$, $\\mathbf{D_i}$, $\\mathbf{E_i}$ from the proof of Theorem~\\ref{thm: perfect decomp} (due to changing $\\delta_1$ to $\\delta_3$ in \\textbf{c)}):\n\\begin{itemize}\n  \\item[$\\mathbf{C_i}$] For each $k\\in [qn_{\\delta}]$ we have $|V_1(i,k)|=(1\\pm \\delta_3)\\delta_3n.$\n  \\item[$\\mathbf{D_i}$] Each $v\\in V(G)$ belongs to at most $2\\delta_3 qn_{\\delta}$ sets among $V_1(i,1),\\ldots, V_1(i,qn_{\\delta})$.\n\\item[$\\mathbf{E_i}$] For any $v\\in V(G)$ and $k\\in [qn_{\\delta}]$ we have $d_{K_{i},V_1(i,k)}(v)\\le \\delta_3^{3\/2}n$.\n\\end{itemize}\n Apply Lemma~\\ref{blowup} to embed $H_{i,k}$ with partition $X_0,X_1,X_2$ and the bijection $\\psi'_{i,k}: X_0\\to V_2(i,k)\\cup V_3(i,k)$ into $W(i,k)$  with $\\delta_3n\/2,2,\\delta_4,\\gamma$ playing the roles of $n,r,\\delta_2,\\gamma$.  As in the proof of Theorem~\\ref{thm: perfect decomp}, $W(i,k)$ is obtained from $W'(i,k)$ by deleting the edges used in previous iterations.\n\nThe verification of the conditions of the blow-up lemma repeats the one done in the previous subsection. \\COMMENT{Indeed, we have the following.\n\\begin{itemize}\n  \\item Inequalities \\eqref{colorbound} and \\eqref{eqcolor}, combined with $\\mathbf {C_i}$ imply that there are at most $n\\cdot 3\\delta_3^2\/4$ edges of colour $c$ in $G[V_1^1(i,k),V_1^2(i,k)]$ (since each of the two parts is roughly a $\\delta_3\/2$-fraction of $V(G)$). Together with   property \\textbf B, it implies that the colouring is $3s\\delta n + 4n\/s+3n\\delta_3^2\/4\\le \\delta_4(\\delta_3n\/2)$-bounded;\n  \\item\\ref{lem blowup 1} is implied by the definition of  by the definitions of $H_{i,k}$ and $X_0$, with $\\Delta=2$;\n\\item \\ref{lem blowup 2} is  implied by the definition of $\\psi'_{i,k}$ and the fact that $|V_1(i,k)|=(1\\pm \\delta_3)\\delta_3n$, while $|V_2(i,k)\\cup V_3(i,k)|\\le \\delta n+2n\/s\\le \\delta_3^2n$;\n\\item \\ref{lem blowup 3} is implied by the fact that $|V_1^{j'}(i,k)|=(1\\pm 2\\delta_3)\\delta_3n\/2$, together with the definition of $H_{i,k}$;\n\n  \\item Due to \\eqref{corr count 5}, the graph $W'(i,k)\\big[V_1^{1}(i,k),V_1^{2}(i,k)\\big]$ is $(\\delta_3,\\gamma d)$-superregular and for each $v\\in V_2(i,k)\\cup V_3(i,k)$, $j'\\in[2]$ we have $d_{W'(i,k),V_1^{j'}(i,k)}(v)=\\gamma d|V_1^{j'}(i,k)|\\pm \\delta_1|V_1^{j'}(i,k)|.$ Let\n      $$K_i^k:=K_i\\cup_{k^*=1}^{k-1}\\psi_{i,k^*}(H_{i,k^*}).$$\n  Then, due to $\\mathbf{E_i}$, for each vertex $v\\in V(G)$ we have $$d_{K_{i}^k,V_1(i,k)}(v)\\le d_{K_{i},V_1(i,k)}(v)+2qn_{\\delta}\\le \\delta_3^{1\/3}\\cdot(\\delta_3n\/2).$$\n  Note that we bounded the contribution of $\\bigcup_{k^*=1}^{k-1}\\psi_{i,k^*}(H_{i,k^*})$ in a trivial way by $2qn_{\\delta}\\le 2n\/g\\le \\delta n$.\n   Using Proposition~\\ref{prop: edge deletion regular}, we conclude that for all $j'\\in[2]$ and $v\\in V_2(i,k)$ $$W(i,k)[V_1^{1}(i,k),V_1^{2}(i,k)]\\text{ is }(\\delta_4,\\gamma d)\\text{-superregular and }\\ d_{W(i,k),V_1^{j'}(i,k)}(v)\\ge \\frac 12\\gamma d|V_1^{j'}(i,k)|.$$\n  Thus,~\\ref{lem blowup 4} and~\\ref{lem blowup 5} are satisfied.\n \\end{itemize}\n}\n\n\n\\section{Concluding remarks}\\label{concl}\nIn this section we describe possible extensions of our results, along with applications. We also adapt a counterexample to Stein's conjecture due to Pokrovskiy and Sudakov \\cite{PS17a} to the setting of Corollary~\\ref{cor1}, as mentioned in the introduction.\n\n\\subsection{Colourings of $K_n$ with no rainbow cycles longer than $\\pmb{ n-\\Omega(\\log n)}$.}\nPokrovskiy and Sudakov \\cite{PS17a} constructed an $n\\times n$ array $A$ where the entries are symbols from $[n]$ such that each symbol occurs precisely $n$ times and such that the largest partial transversal of $A$ has size $n-\\Omega(\\log n)$. Any such array may be interpreted as a colouring of a complete directed graph $G$ on $[n]$ with one loop at each vertex. For any $i,j$, the edge $(i,j)$ of $G$ is coloured with the symbol in the $i$-th row and $j$-th column. In this interpretation, any rainbow directed cycle in $G$ gives raise to a rainbow partial transversal of the same length in $A$.\\COMMENT{But the converse does not hold.} Assuming that $A$ (and thus also the colouring of $G$) is {\\it symmetric}, we can construct a colouring of $K_n$ by simply assigning  the colour of $(i,j)$ and $(j,i)$ to the edge $ij$ in $K_n$, and vice versa. Thus, any rainbow cycle of length $k$ in the resulting $n\/2$-bounded colouring of $K_n$ gives raise to a partial transversal of size $k$ in $A$ (which avoids the diagonal).\n\nAs mentioned in \\cite{PS17a}, the array from \\cite{PS17a} can be easily made symmetric. Moreover, the same argument works for arrays where each symbol appears at most $n-1$ times. This provides us with an example of an $(n-1)\/2$-bounded colouring of $K_n$ whose longest rainbow cycle has length $n-\\Omega(\\log n)$. The array from \\cite{PS17a} can be also adjusted so that the resulting colouring of $K_n$ is locally $n^{1\/2+\\eps}$-bounded for any fixed $\\eps>0$, and so that the length of the longest rainbow cycle is still $n-\\Omega(\\log{n})$, with the constant in the $\\Omega$-term depending on $\\eps$. (In terms of the array, this means that no row or column contains more than $n^{1\/2+\\eps}$ copies of the same symbol.) \\COMMENT{To do so, one has to use only the central squares in the construction from \\cite{PS17a} that have size between $n^{1\/2-\\eps}\\times n^{1\/2-\\eps}$ and $n^{1\/2}\\times n^{1\/2}$ (by terminating the sequence $x_i$ when $x_i\\le n^{1\/2-\\eps}$), and then distribute the colours in the horizontal and vertical boxes so that each of them lies a ``square-like'' rectangle.}\n\n\\subsection{Multipartite versions}\\label{sec:multipartite}\nOur methods also extend to the multipartite setting. We state the following result without proof, as this is almost identical to that of Theorems~\\ref{thm: approx decomp} and~\\ref{thm: perfect decomp}. The two main differences are that we apply a result of MacNeish \\cite{MN} instead of Theorem~\\ref{cl: r-factor} to show that there is a resolvable design in the partite setting. Moreover, in the proof of (i) we apply Lemma~\\ref{counting partite} (with $f$ as in Theorem~\\ref{thm: decomp bip}) instead of Lemma~\\ref{counting quasirandom}. Similarly, to obtain the analogue of \\eqref{corr count 1}, \\eqref{corr count 2} in the proof of (ii), we apply Lemma~\\ref{counting partite} rather than Lemma~\\ref{counting quasirandom}.\\COMMENT{Note that \\eqref{corr count 7} only uses a trivial upper bound, which is also valid in the partite setting.}\n\\begin{theorem}\\label{thm: decomp bip}\nFor given $\\alpha,d_0,f,h,r>0$, there exist $\\eta>0$ and $n_0$ such that the following holds for all $n\\geq n_0$ such that $f$ divides $n$ and $d\\ge d_0$.\nSuppose that $F$ is an $fr$-vertex $h$-edge graph with vertex partition $\\{X_1,\\dots, X_r\\}$ into independent sets of size $f$. Suppose that $a(F)\\le a$.\nSuppose that $G$ is a $rn$-vertex  $r$-partite graph with vertex partition $\\{V_1,\\dots, V_r\\}$ into sets of size $(1\\pm \\eta)n$ and such that $G[V_i,V_j]$ is $(\\eta,d)$-superregular for all $i\\neq j \\in [r]$.\n\\begin{itemize}\n\\item[(i)] If $\\phi$ is a $(1+\\eta) \\frac{f}{h}\\binom{r}{2}dn$-bounded, locally $\\eta n$-bounded colouring of $G$, then $G$ has an $\\alpha$-decomposition into rainbow $\\alpha$-spanning $F$-factors.\n\\item[(ii)] If $\\phi$ is a  $(1-\\alpha) \\frac{f}{h}\\binom{r}{2}dn$-bounded, locally $\\eta n \\log^{-2a}{n}$-bounded colouring of $G$ and $|V_i|=n$ for each $i\\in [r]$, then $G$ has an $\\alpha$-decomposition into rainbow $F$-factors.\n\\end{itemize}\n\\end{theorem}\n\\COMMENT{\n(i) is done in exactly the same way as Theorem~\\ref{thm: approx decomp}: similarly as in the proof of Lemma~\\ref{qr to sr} one can obtain a spanning subgraph $G$ of $G'$ satisfying \\eqref{lem counting 2}--\\eqref{lem counting partite 3}. (One again has to show that the graph formed by the edges violating \\eqref{lem counting partite 3} has small maximum degree.) We removed only a small proportion of the edges, so the graph is still superregular between the parts. Next, we count $R_{G,\\cX,\\cV}(F,v)$, $R_{G,\\cX,\\cV}(F,uw)$ using Lemma~\\ref{counting partite}.\nFinally, we apply Lemma~\\ref{lem: random F decomp} to the rainbow copies of $F$ that respect the partitions $\\cX$, $\\cV$ (that is, contributing to $R_{G,\\cX,\\cV}(F)$). The verification of the conditions is done in the same way.\nIndeed, to check e.g. \\ref{F decomp 2}  note that by\nLemma~\\ref{counting partite} the number of rainbow copies of $F$ containing an edge of a fixed color is at most $(1+\\eta)\\frac fh {r\\choose 2}dn\\cdot (1+\\eps\/3) r_{G,\\cX,\\cV}(F,uw)\\le (1+\\eps)r_{G,\\cX,\\cV}(F,v)$ for any $v,uw$. To see that \\ref{F decomp 3} holds, note that, due to the application of Lemma~\\ref{counting partite} we have $$\\frac{|R_{G,\\cX,\\cV}(F,v)|}{|R_{G,\\cX,\\cV}(F,uw)|}=(1\\pm 3\\eps)\\frac{{r\\choose 2}fdn}{h} = (1\\pm 4\\eps)\\frac {fr|E|}{h|V|}.$$\nNote that $|V(F)|=fr$ so $\\frac {fr|E|}{h|V|}$ really is the right expression here.)\n\nTo prove (ii), we fix prime $b\\sim \\eta^{-1\/2a}\\log n$ and use the result of MacNeish \\cite{MN} to get $b-1$ pairwise orthogonal Latin squares of size $b$. Taking some $r-1$ of them, we can obtain an edge-decomposition of complete $r+1$-partite $K_{r+1}(b)$ into edge-disjoint copies of $K_{r+1}$. (Indeed, if the Latin squares have values $\\ell^1(i,j),\\ldots, \\ell^{r-1}(i,j)$ on the entry $(i,j)$, then we can define the corresponding copy of $K_{r+1}$ by $(i,j,\\ell^1(i,j),\\ldots, \\ell^{r-1}(i,j))$. Then it is easy to see that if any two entries are fixed, then the latter are determined uniquely.) Now, the union of all complete graphs in the decomposition containing vertex $i$ from part $r+1$ forms a spanning factor on the first $r$ parts. Thus, we obtain an edge-decomposition of $K_r(b)$ into edge-disjoint copies of $K_r$, grouped in $b$ groups, each forming a spanning factor. (In other words, we obtain a resolvable $K_r$-decomposition of $K_r(b)$.) Note that this time the number $q$ of $K_r$'s in each factor satisfies $q=b$ (instead of $q=b\/r$ as before).\n\nNext, we split the vertices of each part using a random partition $(1\/b,\\ldots, 1\/b)$. The graphs between the parts are still superregular. Each $K_r=\\{i_1,\\ldots, i_r\\}$ from the decomposition from the previous paragraph corresponds to a $r$-partite graph $G_i^j$ between parts $U^{1}_{i_1},\\ldots, U_{i_r}^r$ (here the upper index refers to the part in $G'$).  We split the colors and get an approximate decomposition of each $G_i^j$ in the same way as in Theorem~\\ref{thm: perfect decomp}. Let us check that \\ref{F decomp 2} holds.\nAn analogue of \\eqref{eqcolor} in our situation states that\n$$e(G[U_{i_1}^1,\\ldots, U_{i_r}^r](\\phi,c)=\\frac 1{b^2}e(G(\\phi,c))\\pm \\frac{\\zeta}{b^2}n.$$\nLemma~\\ref{counting partite} implies that\n$$\\frac{r_{G_i^j}(F,v)}{r_{G_i^j}(F,uw)} =  (1\\pm 2\\eps)\\frac{ fd{r\\choose 2}n\/b}{h}=(1\\pm 3\\eps)\\frac{fr|E(G_i^j)|}{h|V(G_i^j)|}.$$\nFurthermore, $|E(G_i^j(\\phi, I_r))|=(1\\pm \\eps\/2)p_r|E(G_i^j)|$ by\nLemma~\\ref{lem: colour partition}, where $p_r=(1-\\gamma)\/q=(1-\\gamma)\/b$. Using the equalities above and the boundedness of the colouring, for each colour $c\\in I_r$ we have\n$$|E(G_i^j(\\phi,c))|\\le \\frac{(1-\\alpha)\\frac{f{r\\choose 2}dn}{h} \\pm\\zeta n}{b^2}\\le \\frac{(1-2\\gamma)f{r\\choose 2}dn}{h b^2}\\le \\frac{(1-\\gamma)fp_r{r\\choose 2}dn}{bh}\\le \\frac{fr|E(G_i^j(\\phi,I_r))|}{h|V(G_i^j)|}.$$\n(Similarly as for (i), we really want $fr$ instead of $f$ here.)\nThus, for any $v\\in V(G_i^j)$ and $c\\in I_r$, the number of rainbow copies of $F$ in $G_i^j(\\phi,I_r)$ containing  an edge of colour $c$ is at most $$|E(G_i^j(\\phi,c))|\\cdot \\max_{uw\\in E(G_i^j)}\\big\\{r_{G_i^j(\\phi,I_r)}(F,uw)\\big\\}\\le \\frac1{1- 5\\eps} r_{G_i^j(\\phi,I_r)}(F,v),$$ and condition \\ref{F decomp 2} is satisfied.\n\n Everything stays the same until the stage with the definition of $H_{i,k}$. Denote $V_2^j(i,k):=V_2(i,k)\\cap V_j$, $j\\in [r]$ (recall that $V_j$, $j\\in[r]$, are the parts of our initial graph $G$)\nand define  $V_1^j(i,k)$, $V'^{j}(i,k)$ similarly. Thus $\\cD_i(k)\\setminus V_1(i,k)$ is an $F$-factor on $V(G)\\setminus(V_1(i,k)\\cup V_2(i,k))$ and for each $j$ this factor covers precisely all vertices in $V_j\\setminus (V_1^j(i,k)\\cup V_2^j(i,k))$.\nNote that (a) $|V'^{j}(i,k)|=|V'^{j'}(i,k)|$ for all $j,j' \\in [r]$\nand (b) $f \\mid  |V'^{j}(i,k)|$\nand by definition (c) $|V_1^{j}(i,k) \\setminus V'^{j'}(i,k)|=|V_1^{j'}(i,k) \\setminus V'^{j'}(i,k)|$ for all  $j,j' \\in [r]$.\nThus (d)\n$$|V_1^{j}(i,k)|+|V_2^{j}(i,k)|=|V_1^{j}(i,k)\\setminus V'^{j}(i,k)|+|V'^{j}(i,k)|=|V_1^{j'}(i,k)|+|V_2^{j'}(i,k)|=:n'$$\n for all  $j,j' \\in [r]$  and (e) $f \\mid n'$.\n\nDefine\n$H_{i,k}$ to be a disjoint union of $n'\/f$ copies of $F$.\nDefine a partition of $V(H_{i,k})$ into $X_0,\\dots,X_r$ as follows\n(where we will embed $X_j$ into $V_j$ for $j \\in [r]$).\nStart with an equipartition of $V(H_{i,k})$ into $X_1,\\dots,X_r$, so that each of the\n$n'\/f$ copies of $F$ has $f$ vertices in each class.\nNow for each $j\\in[r]$ move $|V^{j}_2(i,k)|$ vertices $v_1^j,\\ldots, v_{t_j}^j$\nfrom $X_j$ into $X_0$. We do this in such a way that no copy of $F$ contains more that $1$ such vertex (this is possible as\n$|V_2^{j}(i,k)| \\le (2 \\delta\/ \\delta_1) |V_1^{j}(i,k)|$).\nNext, we define $\\psi'_{i,k}$ by mapping $v_1^j,\\ldots, v_{t_j}^j$ into $V^j_2(i,k)$.\n\n The rest of the proof is essentially the same\n(note that  we make use of the fact that condition (A5) only needs to hold for those\n$j$ with\n$N_{H_{i,k}}(x) \\cap X_j \\neq \\emptyset$).\n}\n\nNote that, if $F=K_2$, then the above theorem implies that in a properly coloured complete balanced bipartite graph on $2n$ vertices, if no colour appears more than $(1-o(1))n$ times, then we can obtain a $o(1)$-decomposition into rainbow perfect matchings. This was first announced by Montgomery, Pokrovskiy and Sudakov~\\cite{MPS_Harvard}. In terms of arrays, this result states that any $n \\times n$-array filled with symbols, none of which appears more than $(1-o(1))n$ times in total or is repeated in any row or column, can be $o(1)$-decomposed into full transversals. (Note that our theorem has a much weaker condition on the repetitions of symbols in rows or columns.)\n\n\n\n\n\n\n\n\n\n\\subsection{Further remarks and extensions}\nWe can easily deduce the following pancyclicity result from Theorem~\\ref{thm: spanning cycle} and Theorem~\\ref{blowup}:\n  For any $\\eps>0$ there exist $\\eta>0$ and $n_0$ such that whenever $n\\ge n_0$, any $(1-\\eps)\\frac{n}2$-bounded, locally $\\frac{\\eta n}{\\log^{4} n}$-bounded colouring of $K_n$ contains a rainbow cycle of any length.\n  (Indeed, to obtain cycles of length $k$ for $k$ linear in $n$, apply\n  Theorem~\\ref{thm: spanning cycle} to a random subset of $V(G)$ of size $k$,\n  and for shorter cycles, apply Theorem~\\ref{blowup} to $G$.\n \\COMMENT{For $N\\ge \\mu n$ (with $\\eta \\ll \\mu \\ll 1$) it follows by taking a random subset $V'$ of $V(G)$ of size $N$ and finding a Hamilton cycle in $G[V']$ via Theorem~\\ref{thm: spanning cycle}. (The graph is still quasirandom by Lemma~\\ref{qr to sr}.\nMoreover, the coloring has necessary global boundednes conditions due to \\eqref{colorbound} and the application of Lemma~\\ref{lem: graph partition}.\nIndeed the probability of failure is $e^{-\\zeta^3 \\mu n\/ (\\eta n\/\\log n)} =\nn^{-\\zeta^3 \\mu \/\\eta} \\le n^{-3} \\ll 1\/m$ if $\\eta \\ll \\mu, \\zeta$. The initial local boundedness is enough.)\n\nIf $N<\\mu n$, then we can select three random subsets $V_1,V_2,V_3$ of size $\\mu^{2\/3}n$ and apply Theorem~\\ref{blowup} directly with $r=3$, $X_0$ and $V_0$ being empty, and $H$ being a cycle of length $N$. The conditions of the blow-up lemma are obviously satisfied with $\\mu^{1\/3}$ playing the role of $\\delta_2$. Indeed, the colouring is $\\mu^{1\/3}\\cdot \\mu^{2\/3}n$-bounded using Lemma~\\ref{lem: graph partition} and out of the conditions of Lemma~\\ref{blowup} we only need to verify \\ref{lem blowup 3} and \\ref{lem blowup 4}, but both follow from Lemma~\\ref{qr to sr}. Note that we need three sets since odd cycles have chromatic number $3$.\n}\nThis (up to logarithmic factors) extends a result of Frieze and Krivelevich~\\cite{FK08}, who proved this for $\\eta n$-bounded colourings.\n\n\nThe conditions of Theorem~\\ref{thm: approx decomp} (as well as in its bipartite analogue) may be substantially weakened if $F$ is an edge. More precisely, we can prove the following theorem, which applies to {\\it sparse} graphs.\n\n\\begin{theorem}\\label{thm: approx decomp edge}\nFor any $\\delta>0$, there exist $\\eps>0$ and $n_0$ such that the following holds for all $n\\ge n_0$ and $r\\ge \\eps^{-1}$. Suppose that $G$ is an $n$-vertex graph satisfying $d(v)=(1\\pm \\eps)r$. If $\\phi$ is a $(1+\\eps)r$-bounded, locally  $\\eps r$-bounded colouring of $G$, then $G$ contains a $2\\delta$-decomposition into $\\delta$-spanning rainbow matchings.\n\\end{theorem}\n\\begin{proof}\n We apply Lemma~\\ref{lem: random F decomp} with $\\cF$ being a collection of {\\it pairs} of disjoint edges of distinct colour. First of all, let us calculate the values of different subfamilies of $\\cF$ (in the notation of Lemma~\\ref{lem: random F decomp}). For every $v\\in V(G)$ we have $$|\\cF(v)|=(1\\pm \\eps)r\\cdot (1\\pm2\\eps)\\frac {rn}2=(1\\pm 4\\eps)\\frac{r^2n}{2}.$$ Indeed, we first choose an edge adjacent to $v$, and then another edge in $G$ of another colour and disjoint from the first one. The number of edges of $G$ is  $(1\\pm \\eps)\\frac{rn}2$, and the two conditions imposed on the choice of the second edge exclude at most $(3+\\eps)r$ edges. We present the other calculations more concisely. For all edges $uw, u'w'\\in E(G)$, vertices $v_1,v_2\\in V(G)$ and colours $c_1,c_2\\in [m]$ we have\n \\begin{alignat*}{3}\n |\\cF(uw)|\\ &=&&\\ (1\\pm 2\\eps)\\frac{rn}2, && \\\\\n |\\cF(c_1)|\\ &\\le&&\\  (1+\\eps)r\\cdot (1+\\eps)\\frac{rn}2&&\\le (1+3\\eps)\\frac{r^2n}{2},\\\\\n |\\cF(v_1,v_2)|\\ &\\le &&\\ (1+\\eps)\\frac{rn}2+((1+\\eps)r)^2 &&\\le 2rn,\\\\\n |\\cF(c_1,v_1)|\\ &\\le&&\\ \\eps r\\cdot(1+\\eps)\\frac{rn}2+((1+\\eps)r)^2&&\\le 2\\eps r^2n, \\\\\n |\\cF(c_1,c_2)|\\ &\\le &&\\ ((1+\\eps)r)^2&&\\le 2r^2,\\\\\n |\\cF(uw,u'w')|\\ &\\le&&\\ 1.\n \\end{alignat*}\nUsing the displayed formulas and the fact that $r>\\eps^{-1}$, it is easy to see that the conditions of Lemma~\\ref{lem: random F decomp} are satisfied with $8\\eps$ playing the role of $\\eps$.\n\\end{proof}\n\n\n\\noindent {\\bf Postscript.} As mentioned in Section~\\ref{sec:multipartite}, Montgomery, Pokrovskiy and Sudakov had earlier announced the case $F=K_2$ of Theorem~\\ref{thm: decomp bip}(ii) for proper (i.e.~locally 1-bounded) colourings. After completing our manuscript, we learned that they independently obtained some other related results to ours. Slightly more precisely, they also obtained similar approximate decomposition results for rainbow Hamilton cycles, and in addition, they also obtained approximate decomposition results for rainbow trees, but do not consider general rainbow $F$-factors.\nFor their results on rainbow spanning subgraphs, the colourings considered in~\\cite{MPS} are always proper. On the other hand, the global boundedness condition in~\\cite{MPS} is less restrictive than ours. They also deduce from their results a conjecture of Akbari and Alipur on transversals in generalized Latin squares. This conjecture was proved independently by Keevash and Yepremyan~\\cite{KY}. \n\nMore recently, the Brualdi-Hollingsworth conjecture as well its strengthening\nby Constantine has been proved by Glock, K\\\"uhn, Montgomery and \nOsthus~\\cite{GKMO}, i.e.~every sufficiently large optimally edge-coloured complete graph \nhas a decomposition into isomorphic rainbow spanning trees.\nAmongst others, the proof makes use of some of the ideas in the current paper.\nThe related conjecture of Kaneko, Kano, and Suzuki (which allows for \nnot necessarily optimal proper colourings) remains an interesting open problem.\n\n\n\n\n\n\n\n\n\\bibliographystyle{amsplain}\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nUnconventional superconductivity is of great interest both from theoretical as well as experimental points of view \\cite{sigrist1991phenomenological, fulde1964superconductivity}. Anderson's theorem~\\cite{PhysRevB.30.4000} states that in the presence of both time-reversal and inversion symmetries one gets even-parity spin-singlet pairing in superconductors. The absence of either one of these symmetries - either through Zeeman effect (loss of time-reversal symmetry) or spin-orbit interaction (loss of inversion symmetry)  leads to the lifting of spin degeneracy favoring the formation of odd-parity spin-triplet cooper pairs~\\cite{sigrist2009introduction}.  The effect of broken time-reversal symmetry on parity of cooper pairs is pretty well studied. There are several examples of superconductors in nature where the presence of magnetism leads to the appearance of non-trivial pairing - well known examples being heavy Fermion systems (e.g. CeIn$_3$ \\cite{fukazawa2003theory},  CeCoIn$_5$ \\cite{petrovic2001heavy} and UGe$_{2}$ \\cite{huxley2001uge}),  iron-pnictides \\cite{yin2009scanning}, certain organic superconductors. On the other hand, known examples of naturally occurring odd-parity pairing induced by spin-orbit interaction (SOI)  are much rarer - the obvious exceptions being non-centrosymmetric superconductors like CePt$_3$Si \\cite{samokhin2004cept}, CeIrSi$_3$ \\cite{tada2010spin} and CeRhSi$_3$ \\cite{tada2010spin,kimura2007extremely}. Under certain conditions, odd-parity pairing can be induced in two-dimensional superconductors in the presence of SOI \\cite{PhysRevLett.87.037004,michaeli2012superconducting}.\n\nThe quasi-two-dimensional  electron gas (q-2DEG) formed at the interface between (001) oriented SrTiO$_3$ and LaAlO$_3$ (hereafter referred as LaAlO$_3$\/SrTiO$_3$) is one such system. Two factors lead to the appearance of a large Rashba SOI in this system: (a) breaking of parity symmetry at the interface, and (b) a large electric field perpendicular to the interface, primarily due to polar catastrophe (and to a lesser extent due to applied back-gate voltage). It is interesting to note that Rashba SOI has two notable consequences: (a) it induces charge inhomogeneity at the interface at sub-micron length scales~\\cite{caprara2012intrinsic}, and (b) it induces an in-plane field perpendicular to the $k$-vector of the charge carriers \\cite{zhong2013theory}. Both these factors are  expected to have a significant influence on superconductivity. Another  advantage of this q-2DEG over conventional non-centrosymmetric bulk superconductors is that both superconducting $T_C$ and SOI strength are gate-voltage tunable~\\cite{shalom2010tuning, PhysRevLett.104.126803, caviglia2008electric}. A variety of exotic phenomena have been theoretically predicted to exist as a consequence of the SOI including  Fulde-Ferrell-Larkin-Ovchinikov-type (FFLO) superconductivity coexisting with ferromagnetism ~\\cite{michaeli2012superconducting},  exotic superconducting pairing states which are an admixture of spin-singlet and spin-triplet components~\\cite{PhysRevB.92.174531,0034-4885-80-3-036501} and emergent Majorana quasiparticles \\cite{mohanta2014topological}.\n\n\nIn this paper we present detailed experimental studies of the effect of SOI on the magnetotransport and spin fluctuations in high-quality  LaAlO$_3$\/SrTiO$_3$ heterostructures at temperatures much below the superconducting $T_C$. Study of second- as well as higher-order moments of  fluctuations of dynamical variables is a well established tool to probe the presence of long-range correlations in systems undergoing phase transitions~\\cite{weissman1993spin,PhysRevLett.100.180601, koushik2013correlated, samanta2012non, daptary2014probing, daptary2016correlated, daptary2018effect}. From magnetotransport measurements we identify the relevant field scales: upper critical field B$_{c2}$ and spin-orbit field B$_{SO}$, which are gate voltage tunable. We observe that close to these field scales, resistance fluctuations and their higher order statistics develop strikingly non-trivial features. Both from experimental and theoretical data, we find that the interplay between spin-orbit interaction, pairing energy and Zeeman energy creates a fascinating phase diagram  very distinct from that usually found for conventional two-dimensional (2D) superconductors.\\\\\n\n\\section{EXPERIMENTAL DETAILS}\nOur measurements were performed on samples with 10 unit cells of LaAlO$_{3}$ grown by pulsed laser deposition (PLD) on TiO$_{2}$ terminated (001) SrTiO$_{3}$ single crystal substrates. As received SrTiO$_{3}$ substrates were pre-treated with standard buffer hydrofluoric (NH$_4$F - HF) HF solution~\\cite{kawasaki1994atomic} in order to achieve uniform TiO$_{2}$ termination. The TiO$_2$ termination of the substrate realized with the buffer HF solution etching was confirmed from atomic force microscopy measurements. Prior to deposition the treated substrates were annealed for an hour at 830$^\\circ$C in oxygen partial pressure of 7.4 x 10$^{-2}$ mbar. The purpose of pre-annealing of substrates in oxygen atmosphere at 830$^\\circ$C was to remove any moisture and organic contaminants from the surface and also to reconstruct the surface so that pure TiO$_2$ termination is realized. Further, 10 unit cells LaAlO$_{3}$ were deposited at 800$^\\circ$C at an oxygen partial pressure of 1x 10$^{-4}$ mbar.  Growth with the precision of single unit cell was monitored by the oscillations count using in-situ RHEED gun. The epitaxial nature of the films was confirmed by HRXRD performed on a 20 unit-cell LaAlO$_3$ film grown under identical conditions on TiO$_2$ terminated SrTiO$_3$ which allowed us to measure the c-axis lattice parameter of LaAlO$_3$. The thickness of one unit cell from these measurements came out to be 3.75~\\AA~\\cite{kumar2015enhanced}. Ohmic electrical contacts were achieved by  ultrasonically bonding Au wires (25~$\\mu$m diameter) at the four corners of the device in a van der Pauw geometry. This technique is known to breakdown the 10~u.c. of LaAlO$_3$ and provide ohmic contact with the underlying electron gas \\cite{caviglia2008electric, joshua2012universal,han2014two,shalom2010tuning,daptary2016correlated, PhysRevB.95.174502, daptary2018effect}. All electrical measurements were performed in a cryogen-free dilution refrigerator over the temperature range 20--250~mK and magnetic field range 0--16~T. The relative angle between the magnetic field $B$ and the q-2DEG could be changed by rotating the sample \\textit{in-situ} the dilution refrigerator and measurements were done with $B$ applied both parallel ($B_\\parallel$) and perpendicular ($B_\\perp$) to the interface. The charge carrier density at the interface was controlled using a back gate voltage $V_g$ with the SrTiO$_3$ acting as the dielectric material. Measurements were performed over the range $-200$~V$<V_g<$200~V. The interface was found to be superconducting for all values of $V_g>-10$~V.\n\\section{RESULTS AND DISCUSSION}\nWe start with the results of magnetoresistance measurements at $V_g = 200$~V. The superconducting transition temperature $T_C$ (defined as the temperature where the zero field resistance became 40\\% of its normal state value) was measured to be about 140~mK.  Figure~\\ref{fig:figure1}(a) presents the normalized magnetoresistance $R_{sheet}\/R_{sheet}^N$ as a function of perpendicular field $B_\\perp$ at different temperatures for $V_g=200$ V. Here $R_{sheet}^N$ is the zero-field normal-state sheet resistance measured at $T=300$~mK. Fields of the order of 10~mT is enough to destroy the dissipationless superconducting state. The corresponding plots for $B_\\parallel$ are shown in Fig.~\\ref{fig:figure1}(b). As expected, given the quasi-2D nature of the system, the fields required in this case were at least two-orders of magnitude higher. \n\nIn Fig. \\ref{fig:figure1}(c) we plot the upper-critical field $B_{c2}$ (defined as the field at which the $R_{sheet}(B)$ drops to 40~\\% of $R_{sheet}^N$) versus $T$ for both $B_\\perp$  and $B_\\parallel$. The values of $B_{c2}$ have been normalized by the BCS paramagnetic Pauli limit $B_p$, defined as  $\\sqrt{2}g\\mu_BB_p = 3.5k_BT_C$ \\cite{chandrasekhar1962note,clogston1962upper}; $g$ being the gyromagnetic ratio, $k_B$  the Boltzmann constant  and $\\mu_B$ the Bohr magneton. The dependence of $B_{c2}$ on the temperature $T$ for the out-of-plane  is fitted well by the phenomenological 2D Ginzburg-Landau model \\cite{tinkham1963effect}\n\\begin{align}\n\tB_{c2\\perp}=\\frac{\\Phi_0}{2\\pi \\xi_{GL}(0)^2}(1-T\/T_c)\n\t\\label{Eqn:GL1}\n\\end{align}\n\nwhere $\\xi_{GL}(0)$ is the in-plane GL coherence length at $T=0$ K, $\\Phi_0=h\/2e$ is the flux quantum. The value of $\\xi_{GL}(0)$ extracted from the fit is 55~nm which matches well with previous reports \\cite{reyren2009anisotropy,shalom2010tuning}. From Fig. \\ref{fig:figure1}(c) we observe that $B_{c2\\parallel}$ far exceeds the Clogston-Chandrashekhar limit which, in the weak coupling approximation, is expected to limit the value of the parallel upper critical field to $B_{c2\\parallel} \\leq B_p$. This large enhancement of  $B_{c2\\parallel}$ has been reported previously in (001) LaAlO$_3$\/SrTiO$_3$ hetero-interfaces~\\cite{shalom2010tuning}  and has been postulated to arise from the presence of strong Rashba SOI which weakens spin paramagnetism by mixing the quasiparticle  spin states~\\cite{PhysRevB.12.877, PhysRevB.25.171}. Other possible mechanisms like anisotropic pairing mechanism, strong-coupling superconductivity or other exotic many-body effects have been considered and ruled out by previous workers (see for example~\\cite{PhysRevLett.119.237002, shalom2010tuning}). \n\nFor the case of strong SOI, $B_{c2\\parallel}$ is related to the spin-orbit scattering time through~\\cite{PhysRevB.12.877}:\n\\begin{eqnarray}\n\t\\tau_{SO}=0.362\\frac{\\hbar}{k_BT_c}\\Big(\\frac{B_P}{B_{c2\\parallel}(0)}\\Big)^2\n\t\\label{Eqn:comparison}\n\\end{eqnarray}\nUsing this relation yields $\\tau_{SO}=4\\times 10^{-13}$~s for $V_g$=170~V. \n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width= 0.48\\textwidth]{figure1.pdf}\n\t\t\\small{\\caption{Normalized sheet resistance versus temperature $T$ as a function of (a) perpendicular field, $B_\\perp$ and (b) parallel magnetic field, $B_\\parallel$. (c) Upper critical field $B_{c2}$ normalized by the Pauli paramagnetic field $B_p$ as a function of reduced temperature $T\/T_C$ for fields applied parallel to the interface (blue filled circles) and perpendicular to the interface (red open circles). The gray dotted lines are fits to Eqn.~\\ref{Eqn:GL1}.\n\t\t\t\tThe measurements were performed at $V_g=200$~V. \n\t\t\t\t\\label{fig:figure1}}}\n\t\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width= 0.48\\textwidth]{figure2.pdf}\n\t\t\\small{\\caption{(a) Magnetoconductance as a function of $B_\\perp$ at different values of $V_g$. The scatter points are the measured data points while the solid lines are fits to the  Eqn.~\\ref{Eqn:WAL}. (b) Plot of $B_{SO}$ (olive filled circles) and  $B_{c2\\parallel}$ (red filled circles) versus $V_g$.  The measurements were performed at 245~mK. \\label{fig:figure2}}}\n\t\\end{center}\n\\end{figure}\n\nThe  SOI strength can also be  extracted from the measured low-field magnetoconductance at $T>T_C$. In a two dimensional system with in-plane SOI, in the presence of a perpendicular magnetic field $B_\\perp$, the correction to conductance $\\Delta \\sigma$ takes the Maekawa-Fukuyama  form~\\cite{maekawa1981magnetoresistance}:\n\\begin{align}\n\t\\Delta\\sigma(B)= &\\frac{e^2}{\\pi h}\\Big[\\Psi\\Big(\\frac{B_\\perp}{B_i+B_{SO}}\\Big) \\notag \\\\\n\t&+\\frac{1}{2\\sqrt {1-\\gamma^2}} \\Psi\\Big(\\frac{B_\\perp}{B_i+B_{SO}(1+\\sqrt{1-\\gamma^2}}\\Big) \\notag \\\\\n\t&-\\frac{1}{2\\sqrt {1-\\gamma^2}} \\Psi\\Big(\\frac{B_\\perp}{B_i+B_{SO}(1-\\sqrt{1-\\gamma^2}}\\Big)\\Big].\n\t\\label{Eqn:WAL}\n\\end{align}\nHere $\\Psi(x)$ = ln(x)+$\\psi(0.5+\\frac{1}{x})$, where $\\psi$ is the digamma function. $B_i = \\hbar\/(4eD\\tau_i)$ and $B_{SO} = \\hbar\/(4eD\\tau_{SO})$ are inelastic and spin-orbit fields respectively ($\\tau_i$ and $\\tau_{SO}$ are respectively the inelastic and spin-orbit scattering times), $D$ is the diffusion constant and $\\gamma$ is the Zeeman correction $\\gamma=g\\mu_B B\/4eDB_{SO}$ ($g$ and $\\mu_B$ are the electron $g$ factor and Bohr magnetron respectively).\n\nThe low-field magnetoconductance at $T=245$~mK is plotted in Fig.~\\ref{fig:figure2}(a).   From the fits to these curves we extract the $\\tau_{SO}$ and  $B_{SO}$. The value of $\\tau_{SO}$ extracted from the fits to the magnetoresistance measured at $V_g$=170~V is $1.6\\times10^{-13}$~s which matches closely with the value extracted using Eqn.~\\ref{Eqn:comparison}. The value of $\\tau_{SO}$, $\\tau_{i}$ and $\\tau_{elas}$ (elastic scattering time) are shown in Fig. \\ref{fig:appendix1} in Appendix. As shown in Fig.~\\ref{fig:figure2}(b), the value of $B_{SO}$ increases by almost two orders of magnitude as $V_g$ is swept from -200~V to 200~V. At low  $V_g$, $B_{SO}$ and $B_{c2\\parallel}$ are comparable (Fig.~\\ref{fig:figure2}(b)). With increasing $V_g$, $B_{SO}$ increases rapidly and becomes significantly larger than $B_{c2\\parallel}$.  \n\nTo probe the effect of spin-orbit interactions on charge carrier dynamics in the superconducting state, we studied resistance fluctuations for different  magnetic fields at $T=20$~mK ($T\/T_C\\approx 0.1$). The measurements were performed using a standard four-probe ac measurement technique (For details see Ref.~\\cite{ghosh2004set}). Briefly, at each value of $V_g$ and $B$, the device is biased by a small ac current and the time series of resistance fluctuations $\\delta R_{sheet}(t)$ is measured for 30 min using a dual-phase digital lock-in amplifier. The output of the lock-in amplifier is recorded by a fast data acquisition (DAQ) card. After extensive digital filtering of $\\delta R_{sheet}(t)$ to remove line frequency and aliasing-effects, the power spectral density (PSD) of resistance fluctuations $S_R(f)$ was calculated using the method of Welch Periodogram.  The time-series of resistance fluctuations for a few representative values of $B_\\parallel$, measured at $T$=20~mK and $V_g$=200~V, are plotted in Fig.~\\ref{fig:figure3}(a). The corresponding PSD are shown in Fig.~\\ref{fig:figure3}(b). For all values of $V_g$ and $B$, the dependence of $S_R(f )$ on the frequency $f$ was  found to be of the form $S_R(f) \\propto 1\/f^\\alpha$ with $\\alpha \\sim 0.9-1$. $S_V(f)$ was always found to depend quadratically on the voltage $V$ developed across the channel [see inset of Fig~\\ref{fig:figure3}(b)] establishing that the measured noise  originated from resistance fluctuations of the sample.\n\nThe PSD of resistance fluctuations was integrated over the measurement bandwidth (7~mHz-4~Hz) to obtain the relative variance of resistance fluctuations $\\mathcal{R}$:\n\\begin{align}\n\t\\mathcal{R} \\equiv \\frac{\\langle \\delta R_{sheet}^2 \\rangle}{\\langle R_{sheet}^2\\rangle}=\\frac{1}{\\langle R_{sheet}^2\\rangle}\\int S_R(f)df \n\t\\label{PSD}\n\\end{align}\nIn Fig. \\ref{fig:figure4}(a) we show the plots of relative variance of resistance fluctuations $\\mathcal{R}$ as a function of $B_\\parallel$ at a few representative values of $V_g$ at $T=20$~mK. At high $B_\\parallel$, the noise has a very shallow dependence on the field. Below a certain characteristic field, which is specific to $V_g$, the noise increases rapidly with decreasing $B$. Normally, one would expect this characteristic field to be the upper critical field, above which superconducting fluctuations are suppressed. However, a closer inspection of the data reveals that the characteristic field scale in this case is the spin-orbit field $B_{SO}$. As the field decreases below $B_{SO}$, the noise increases rapidly -- growing by over four orders of magnitude in the narrow magnetic field range $B_{c2\\parallel}<B_\\parallel<B_{SO}$.\nIn Fig.~\\ref{fig:figure4}(b) we show a scaling plot of the noise $\\mathcal{R}(B)\/\\mathcal{R}(B_{SO})$ as a function of $B\/B_{SO}$. The data for all $V_g>-10$~V collapse onto  a single curve showing that indeed $B_{SO}$ is the relevant scale governing the $B_\\parallel$ dependence of the resistance fluctuations in a superconductor with strong SOI. \n\nTo understand the origin of the measured resistance fluctuations, we studied their higher-order statistics. Such studies have been used extensively to detect the presence of long-range correlations in systems undergoing magnetic, spin-glass or superconducting transitions~\\cite{weissman1993spin, PhysRevLett.100.180601, koushik2013correlated, samanta2012non, daptary2014probing, daptary2016correlated, daptary2018effect}. The Central Limit Theorem states that for uncorrelated random fluctuators, the fluctuation statistics is Gaussian. As the correlation length in the system begins to diverge  near a critical phase transition, the resultant time-dependent fluctuation statistics becomes strongly non-Gaussian~\\cite{weissman1993spin, PhysRevLett.100.180601, koushik2013correlated, daptary2018effect}.  We computed the `second spectrum'  which is the four-point correlation function of the resistance fluctuations over a chosen frequency octave ($f_l, f_h$)~\\cite{PhysRevB.31.2254, PhysRevB.53.9753}. It is mathematically defined as\n\\begin{equation}\n\tS_R^{f_1}(f_2)=\\int_0^\\infty \\langle\\delta R^2(t)\\rangle\\langle\\delta R^2(t+\\tau)\\rangle \\cos(2\\pi f_2\\tau)d\\tau\n\t\\label{SP}\n\\end{equation}\nwhere $f_1$ is the center-frequency of the chosen octave and $f_2$ the spectral frequency. Physically, $S_R^{f_1}(f_2)$ represents `spectral wandering' of the PSD with time. To avoid corruption of the signal by the Gaussian background noise, the second spectrum was calculated over the frequency octave 93.75--187.5 mHz, where the sample noise is significantly higher than the background noise. A convenient way of representing the second spectrum is through its normalized form $S_N^{(2)}$ defined as\n\\begin{equation}\n\tS_N^{(2)}=\\int_0^{f_h-f_l}S_R^{f_1}(f_2)df_2\/[\\int_{f_l}^{f_h}S_R(f)df]^2\n\t\\label{NSP}\n\\end{equation}\nFor Gaussian fluctuations, $S_N^{(2)}$ = 3. The measured values of $S_N^{(2)}$  as a function of $B_\\parallel$ is shown in Fig. \\ref{fig:figure5}(a). We see that as the magnetic field is decreased below $B_{c2\\parallel}$, $S_N^{(2)}$ starts increasing monotonically from its high field value which was close to 3. This can be appreciated better from Fig.~\\ref{fig:figure5}(b) where we plot $S_N^{(2)}(B_\\parallel)\/S_N^{(2)}(B_{c2\\parallel})$ as a function of $B_\\parallel\/B_{c2\\parallel}$. The data for all $V_g$ collapse onto a single plot showing  that the relevant field scale for the second spectrum is $B_{c2\\parallel}$. We note that scaling plot $S_N^{(2)}(B_\\parallel)\/S_N^{(2)}(B_{c2\\parallel})$ remains unchanged if $B_{c2\\parallel}$ are defined for other resistance criterion, e.g., $R_{sheet} = 0.7R_{sheet}^N$ and $R_{sheet} = 0.1R_{sheet}^N$ (see Fig. \\ref{fig:appendix2} ). For $V_g= -10$ V, where the device is in resistive state over the entire magnetic field range, the relative variance of resistance fluctuations $\\mathcal{R}$ is independent of  field (Fig.~\\ref{fig:figure4}) and $S_N^{(2)}\\simeq3$ (Fig.~\\ref{fig:figure5}) showing that the fluctuations in normal state in LaAlO$_3$\/SrTiO$_3$ are Gaussian. \\\\\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\textwidth]{figure3.pdf}\n\t\t\\small{\\caption{(a) Time series of resistance fluctuations at a few representative values of $B_\\parallel$. The measurement was performed at $T=20$ mK and $V_g=200$~V. (b) PSD of resistance fluctuations corresponding to the time-series shown in (a). Inset: PSD is plotted as a function of $V^2$ at $B=1$ Tesla and $V_g=170$ V - the linear dependence of $S_V(V)$ on $V^2$ establishes that noise originates from resistance fluctuations of the sample. \\label{fig:figure3}}}\n\t\\end{center}\n\\end{figure}\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\textwidth]{figure4.pdf}\n\t\t\\small{\\caption{(a) Plots of relative variance of resistance fluctuations $\\mathcal{R}(B)$ versus $B_\\parallel$, at different values of $V_g$. The measurements were performed at 20~mK. (b) Scaling plot of noise $\\mathcal{R}(B)\/\\mathcal{R}(B_{SO})$ versus $B_\\parallel\/B_{SO}$ for V$_g$= 170 V, 100 V, 50 V, 15 V and -10 V respectively. Also, plotted is  $\\mathcal{R}(B)\/\\mathcal{R}(B_{SO})$ versus $B_\\perp\/B_{SO}$ for field applied perpendicular to the interface at $V_g=200$ V (magenta open circles).  \\label{fig:figure4}}}\n\t\\end{center}\n\\end{figure}\n\n\n\nTo summarize our observations so far: (a) for $B_\\parallel>B_{SO}$, the resistance fluctuations are almost independent of $B_\\parallel$  and have a Gaussian distribution, (b) there is a significant range of field $B_{SO}>B_\\parallel>B_{c2\\parallel}$ where the resistance fluctuations  depend strongly on $B_\\parallel$ while remaining Gaussian, and (c) for $B_\\parallel < B_{c2\\parallel}$, the resistance fluctuations are large, have a strong $B_\\parallel$-dependence and have a non-Gaussian distribution. In the inset of Fig.~\\ref{fig:figure5}(b) we summarize the data.  One can see that the magnetic field at which the second spectrum deviates from the Gaussian value (we call it $B_{NG}$) closely follows the upper critical field $B_{c2 \\parallel}$ while the field at which the noise begins to shoot up (labeled $B_N$) tracks $B_{SO}$. At this point it is profitable to compare these observations with what is seen for $B_\\perp$ for this q-2DEG  superconductor --  a representative data taken at $T$ = 20~mK and $V_g$ = 200~V is plotted in Fig.~\\ref{fig:figure4}(b) (magenta open circles).  For $B_\\perp>B_{c2\\perp}$, the noise is field-independent, small in magnitude and Gaussian. For $B_\\perp<B_{c2\\perp}$,  the noise is non-Gaussian and diverges strongly as the field is reduced ( Fig. \\ref{fig:figure5}(b) - dark yellow squares). Notably, in contrast to $B_\\parallel$, the divergence of noise and appearance of non-Gaussian component are concurrent. This has been observed previously in other 2-dimensional superconductors and has been shown to arise due to long range correlations between the vortices near the transition \\cite{PhysRevLett.76.2551,PhysRevLett.78.519, PhysRevB.57.6036, Wagenblast1998, daptary2016correlated, koushik2013correlated}. \n\\begin{figure}[t!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\textwidth]{figure5.pdf}\n\t\t\\small{\\caption{ (a) Plot of $S_N^{(2)}$ as a function of magnetic field applied parallel to the interface at different values of gate voltages. The measurement was performed at 20 mK. (b) Scaling plot of $S_N^{(2)}(B_\\parallel)\/S_N^{(2)}(B_{c2\\parallel})$ versus $B_\\parallel\/B_{c2\\parallel}$ for $V_g$=170 V, 100 V, 50 V and 15 V respectively.  Also, plotted is  $S_N^{(2)}(B_\\perp)\/S_N^{(2)}(B_{c2\\perp})$ versus $B_\\perp\/B_{c2\\perp}$ for field applied perpendicular to the interface at $V_g=200$ V (dark yellow squares). \\label{fig:figure5}}}\n\t\\end{center}\n\\end{figure}\n\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\textwidth]{appendix2.pdf}\n\t\t\\small{\\caption{ Scaling plot of $S_N^{(2)}(B)\/S_N^{(2)}(B_{c2\\parallel})$ versus $B_\\parallel\/B_{c2 \\parallel}$. $B_{c2}$ are defined from the resistance transition at (a) $0.7R_{sheet}^N$, (b) $0.4R_{sheet}^N$ and (c) $0.1R_{sheet}^N$ resistance criterion.  \\label{fig:appendix2}}}\n\t\\end{center}\n\\end{figure}\n\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\textwidth]{Phase_diagram.pdf}\n\t\t\\small{\\caption{ Schematic phase diagram showing the spin orientations at the  LaAlO$_3$\/SrTiO$_3$ interface.  Upper critical field B$_{c2 \\parallel}$ (blue line) and spin-orbit field B$_{SO}$ (red line) have been plotted in the SOI-energy $\\epsilon_{SOI}$ and $B_\\parallel$ plane. Arrows indicate the direction of spin of a single electron in the plane of the q-2DEG while circles represent the cooper pairs.  \\label{fig:figure6}}} \n\t\\end{center}\n\\end{figure}\n\nWe now discuss the possible origin of the decoupling of $B_{NG}$ and $B_N$ in this system. As shown before~\\cite{koushik2013correlated, daptary2016correlated}, it is correlations between vortices  that leads to non-Gaussian noise in 2D-superconductors. Thus, it is natural that $B_{c2 \\parallel}$  (the field at which superconductivity is destroyed)  and $B_{NG}$ (the field at which non-Gaussian fluctuations vanish) coincide. The measured resistance fluctuations, however, persist beyond $B_{c2 \\parallel}$ deep into the normal state, until $B_\\parallel\\sim B_{SO}$. Below we present a plausible scenario which explains this. Strong SOI present in this system ensures that the electronic spins are all in-plane. As the electronic transport is diffusive, the $k$-vector of the charge carriers take random values. Spin-momentum locking due to SOI causes these charge carriers to feel an effective in-plane $B_{SO}$ field perpendicular to the $k$-vector. The competition of this random $B_{SO}$ with $B_\\parallel$  brings down the in-plane spin magnetic moment to $\\sim( B_\\parallel\/B_{SO})\\mu_B$~\\cite{xi2016ising, youn2012role, sigrist2009introduction}. At large enough parallel magnetic fields, Zeeman energy ensures that all the spins are aligned along $B_\\parallel$. As $B_\\parallel$ is reduced to the order of $B_{SO}$ there begins to appear spins of opposite signs which can form Cooper pairs. With further reduction of $B_\\parallel$, the superfluid density grows and for  $B_\\parallel <B_{c2\\parallel}$ global phase coherence sets in. Thus, in the field range $B_{SO}>B_\\parallel>B_{c2\\parallel}$ there will exist domains of superconducting clusters in a background of normal carriers. We propose that it is fluctuations of these superconducting clusters that give rise to the large Gaussian  noise over this field regime. We present a schematic phase diagram of the spin orientation in Fig.~\\ref{fig:figure6} in the SOI-energy $\\varepsilon_{SOI}$ and $B_\\parallel$ plane. The values of $\\varepsilon_{SOI}$ have been obtained from $\\tau_{SO}$ extracted from the fits to the magnetoresistance data at different $V_g$ using Eqn.~\\ref{Eqn:WAL}. This picture is in some sense analogous to what one gets in the zero-field limit - as the temperature is reduced sufficiently close to $T_C$, there appears percolating clusters with finite superfluid density in a resistive background which gives rise to large Gaussian resistance fluctuations. It has been predicted that FFLO state is favorable in the phases between $B_{c2}$ and $B_{SO}$ \\cite{barzykin2002inhomogeneous,dimitrova2007theory}, which possibly, can have contributions to the resistance fluctuations. Without experimental data, we refrain commenting on it.\n\n The magnetic field-induced transition to the superconducting state is affected by non-magnetic disorder \\cite{mohanta2014oxygen, mohanta2013phase} and the transition is assumed to be percolative in nature. To describe such a percolative phase transition induced by in-plane magnetic field $B$, a random resistor network (RRN) model was considered \\cite{PhysRevLett.54.1718,PhysRevLett.54.2529}. In this model, we consider a square network of identical resistors of size $L \\times L$, where $L$ is the number of grid points along $x$ or $y$ direction. In the ideal scenario, the resistor network is assumed to be connected by external conducting wires to a voltage source $V$, which causes a current $I$ to flow through the network, so that the macroscopic sheet-resistance is measured as $R_{sheet} = V\/I$. In this model, we discretize the resistance and define the mean resistance at a grid point ($x_i,y_i$) by $R_i$, so that the macroscopic resistance is given by averaging over all grid points in the network \\textit{viz.} $R_{sheeet}=(1\/L^2)\\sum_i R_i$.\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\textwidth]{patch_normR_thory.pdf}\n\t\t\\small{\\caption{ Colormap of the sheet-resistance $R_{sheet}$ in a $100\\times100$ network  in the RRN model at different in-plane magnetic fields (a) $B_\\parallel=1$ T, (b) $B_\\parallel=1.3$ T, (c) $B_\\parallel=1.6$ T and (d) $B_\\parallel=1.9$ T, across the transition from superconducting state to the normal metallic state. Blue background denotes regions with resistance $R_{sheet}=0$, while red dots\/patches denote regions with high resistance $R_{sheet}=R_{sheet}^N$ ($R_{sheet}=R_{sheet}^N$ being the resistance in the normal metallic state). In this plot, temperature $T=20$~mK and gate-voltage $V_g=170$~V. (e) The blue-line shows the variation of the normalized resistance $R_{sheet}\/R_{sheet}^N$ (blue curve) with in-plane magnetic field $B_\\parallel$ at  $T=20$~mK and  $V_g=170$~V, where $R_{sheet}^N$ is the resistance in the normal metallic state. The theoretical fit obtained using the RRN model is shown by the red open circles. \\label{fig:Res_map}}}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.48\\textwidth]{tau_dist.pdf}\n\t\t\\small{\\caption{Distribution of the relaxation time at different in-plane magnetic field $B_\\parallel$ values (a) 1 T, (b) 1.3 T, (c) 1.6 T and (d) 1.9 T. The distribution changes from non-Gaussian type to Gaussian type as $B_\\parallel$ is increased across the transition from superconducting state to normal-metallic state. In this plot,  $T=20$~mK and  $V_g=170$~V. \\label{fig:tau_dist}}}\n\t\\end{center}\n\\end{figure}\n\nIn the RRN model, we consider a $100\\times100$ network in which circular resistive clusters appear in the superconducting phase when $B$ is increased, as shown in Fig. \\ref{fig:Res_map}. The $B$-dependence of the number and diameter of the clusters are given, respectively, by $N_{cluster}=Int.(C_1(B-B_c))$ and $D_{cluster}=C_2B_r$, where $B_r=(B-B_c)\/B_c$, $B_c$ is the critical field for the superconducting transition (at a given value of gate voltage $V_g$, we take $B_c$ as the highest available critical field $i.e.$ $B_c=B_{so}$), $C_1$ and $C_2$ are parameters which are determined by fitting $R$ with experimental data, the function $Int.()$ returns the integer value of the number inside the bracket. The value of the resistance inside the resistive clusters is large, here we assume $R_{sheet}=R_{sheet}^N$, the value in the normal metallic state at $B=2$~T. The normalized resistance at a field $B$ is given by $R\/R_N=1\/(1+\\xi^2)$, where $\\xi$ is the superconducting coherence length. We assume that in a disordered BCS superconductor with percolative superconducting transition, $\\xi$ follows a field dependence which is similar to the temperature-dependence, predicted by Halperin-Nelson equation, and can be expressed as $\\xi=(2\/A)\\sinh(b\/\\sqrt{B_r})$, where $A$ and $b$ are parameters which are determined by fitting with experimental data.  By fitting the experimental data at $T=20$~mK and $V_g=170$~V, we obtain $A=1.8$, $b=0.2$, $C_1=1000$ and $C_2=1.65$. The data have been plotted in Fig. \\ref{fig:Res_map}(e). Spatial inhomogeneity on the two-dimensional superconductor broadens the BKT transition \\cite{PhysRevB.80.214506, daptary2016correlated} and a percolation transition is well accessible within the Halperin-Nelson theory.\n\nThe resistance at position ($x_i,y_i$) at a magnetic field $B$ and time $t$ is given by $R_i(B,t)=R_i(B)+\\delta R_i(B,t)$. We start at $B=0.2$~T with $\\delta R_i(B,t=0)=0$ and continuously update $R_i(B,t)$ at the interval of a relaxation time $\\tau$ and finally reach the maximum field $B=2$~T. The amplitude of noise $\\delta R_i(B,t)$ is chosen randomly from a set $\\{ \\delta R_i(B,t) \\}$ of numbers which follows Gaussian distribution and has a standard deviation $0.001$ and zero mean. \nThe statistics of the noise is, however, governed by the distribution of $\\tau$ which is also chosen randomly from a set $\\{ \\tau_n \\}$. We assume that the Josephson junctions, formed during the percolative superconducting transition, contribute non-Gaussian component in the resistance noise. We, therefore, consider that the distribution of the relaxation time has two components which can be expressed as $\\{ \\tau_n \\}=x\\{ \\tau_n \\}_{NGC}+(1-x)\\{ \\tau_n \\}_{GC}$, $NGC$ stands for non-Gaussian component and $GC$ for Gaussian component. The fraction $x$, which defines the amount of non-Gaussianity in the noise, is taken to be proportional to the ratio of the superconducting region to the non-superconducting region. The distribution functions for $\\{ \\tau_n \\}_{NGC}$ and $\\{\\tau_n \\}_{GC}$ are determined by comparing the frequency-dependence of power spectral density (PSD) of resistance noise, given by the following equation, with the experimentally-obtained PSD:\n\\begin{equation}\n\tS_R(f)=\\lim_{t_0\\to\\infty} \\Big( \\frac{1}{2t_0} \\Big) \\Big( \\int_{-t_0}^{t_0} \\delta R(t)e^{i2\\pi ft}dt \\Big)^2\n\\end{equation}\nTo incorporate the $1\/f$-dependence of PSD and the influence of SOI, we include the second critical field $B_{c2||}$ in the PSD, through the following relation:\n\\begin{equation}\n\tS_R(f)=\\int_{0}^{\\infty} d\\tau F(\\tau) \\frac{2\\tau (B-B_{c2||})^3}{1+2\\pi f\\tau},\n\\end{equation}\nwhere $F(\\tau)$ is the distribution function for $\\tau$. For the GC, we have a Gaussian distribution $F(\\tau)=1\/(\\sqrt{2\\pi \\sigma^2})e^{-(\\tau-\\tau_{GC})^2\/2\\sigma^2}$, where $\\sigma$ and $\\tau_{GC}$ are, respectively, the variance and mean value of the Gaussian distribution. For the NGC, we use  a stretched exponential function $F(\\tau) = 1\/(2\\sqrt{\\pi}) \\sqrt{\\tau} e^{-\\tau\/\\tau_{NGC}}$, typically used to study glassy dynamics. With $\\tau_{GC}=\\tau_{NGC}=500$~ns and $\\sigma=100$~ns, the PSDs are calculated at different fields and the corresponding distributions of $\\{ \\tau_n \\}$ are shown in Fig. \\ref{fig:tau_dist}.\n\n\n\n\\begin{figure}[t!]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.5\\textwidth]{noise_phasedia_theory.pdf}\n\t\t\\small{\\caption{ Gate-voltage variation of (a)  $\\mathcal{R}$ with magnetic field $B$, (b) normalized  $\\mathcal{R}$ with $B\/B_{so}$, (c) $S_N^{(2)}$ with magnetic field $B$ and (d) normalized $S_N^{(2)}$ with $B\/B_{c2||}$ at different values of gate-voltage $V_g$. The normalization of the quantities, plotted on the vertical axis, in (b) and (d) is performed using the respective values at the maximum value of the $B$ field. In this plot, temperature $T=20$~mK. (e) Variation of the critical fields $B_{c2||}$ (yellow squres) and $B_{so}$ (blue traingles) with gate-voltage. The modeled gate-voltage dependence is obtained from Fig.\\ref{fig:figure6}. \\label{fig:Noise_B}}}\n\t\\end{center}\n\\end{figure}\nThe relative variance of the resistance fluctuations $\\mathcal{R} \\equiv \\frac{\\langle \\delta R_{sheet}^2 \\rangle}{\\langle R_{sheet}^2 \\rangle}$ and the normalized second spectrum $S_{N}^{(2)}$ are calculated by using Eq. \\ref{PSD} and Eq. \\ref{NSP} respectively. A plot of $\\mathcal{R}$ and $S_{N}^{(2)}$ as a function of the field $B$ for different representative values of gate voltage $V_g$ are shown in Fig. \\ref{fig:Noise_B}(a) and (c). The same set of obtained data, when plotted with respect to the field values, scaled using the critical fields $B_{so}$ and $B_{c2||}$, reveals that  $\\mathcal{R}$ scales with $B_{so}$ while $S_{N}^{(2)}$ scales with $B_{c2||}$, as shown in Fig. \\ref{fig:Noise_B}(b) and (d). The critical fields $B_{c2||}$ (yellow squres) and spin-orbit fields $B_{so}$ (blue traingles), obtained from the simulation are shown as a function of $V_g$ along with experiment  in Fig. \\ref{fig:Noise_B}(e). The excellent match between experimental and simulation data tells that a simple random resistor network model is able to capture the essential features of resistance fluctuations close to the upper critical field in 2D inversion symmetry broken superconductors.\\\\\n\\section{CONCLUSION}\nTo conclude, we have probed, through careful measurements of  resistance fluctuations, the interplay of SOI, pairing potential  and Zeeman energy in the superconducting phase of LaAlO$_3$\/SrTiO$_3$. We find the presence of larger non-Gaussian fluctuations below $B_{c2\\parallel}$ arising due to correlated vortex-dynamics. Large, Gaussian resistance fluctuations were seen in the field range between  $B_{c2\\parallel}$  and $B_{SO}$ which indicate the presence of superconducting clusters  without global phase coherence. We identify and quantify the relevant energy scales in this system -  SOI, Zeeman energy and pairing potential. Our work emphasizes the important role played by the interplay between  these energy scales in   framing the phase diagram of 2-D inversion asymmetric superconductors.\n\\section*{ACKNOWLEDGMENTS}\nThe authors thank R C Budhani, IIT Kanpur for \tproviding the samples. AB acknowledges funding from SERB, DST, Govt. of India.  HKK acknowledges funding from CSIR, Govt. of India. \n\\section*{APPENDIX}\nIn Fig. \\ref{fig:appendix1}, we plot the different scattering times extracted from Eq. \\ref{Eqn:WAL} as a function of $V_g$ at $T=245$ mK. It can be seen that for all $V_g$, total scattering time $\\tau$ (=$\\tau_i$+$\\tau_{elas}$, where $\\tau_i$ and $\\tau_{elas}$ are inelastic and elastic scattering time respectively) is larger than spin-orbit scattering time $\\tau_{SO}$ implying strong spin-orbit interaction in the LaAlO$_3$\/SrTiO$_3$ interface which are gate voltage tunable.\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=0.42\\textwidth]{appendix1.pdf}\n\\small{\\caption{ Plot of spin-orbit scattering time $\\tau_{SO}$ (olive filled circles), inelastic time $\\tau_{i}$ (red filled squares), elastic time $\\tau_{elas}$ (black filled) and total scattering time $\\tau=\\tau_{i}+\\tau_{elas}$ (blue open triangles) versus $V_g$. \\label{fig:appendix1}}}\n\\end{center}\n\\end{figure}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}