diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzcuio" "b/data_all_eng_slimpj/shuffled/split2/finalzzcuio" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzcuio" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe \\emph{$s$-independence number} of a graph $G$, denoted $\\alpha_s(G)$, is the maximum number of vertices in a $K_s$-free induced subgraph of $G$ (so the standard independence number is the same as the $2$-independence number). For a given graph $H$, the \\emph{Ramsey-Tur\\'an number} $\\textup{\\textbf{RT}}_s(n, H, f)$ is the maximum number of edges in any $H$-free graph $G$ on $n$ vertices with $\\alpha_s(G) < f$. If there does not exist any $H$-free graph $G$ on $n$ vertices with $\\alpha_s(G) < f$ then we put $\\textup{\\textbf{RT}}_s(n,H,f)=0$. Observe that the lower bound $k \\le \\textup{\\textbf{RT}}_s(n,H,f)$ means that there exists an $H$-free graph~$G$ of order~$n$ with $\\alpha_s(G) < f$ and at least $k$ edges. The upper bound $\\textup{\\textbf{RT}}_s(n,H,f) < \\ell$ says that there is no $H$-free graph~$G$ of order $n$, $\\alpha_s(G) < f$, and at least $\\ell$ edges.\n\nIn general it is far from trivial to even determine the existence of any $H$-free graph~$G$ on $n$ vertices with $\\alpha_s(G) 0$ and an integer $r\\ge 2$\n\\begin{equation}\\label{f:r}\n\\Omega\\big{(}n^{\\frac{1}{2} - \\varepsilon}\\big{)} = \\textup{\\textbf{f}}_{s,s+r}(n) = O(\\sqrt{n})\n\\end{equation}\nfor all sufficiently large~$s$.\n\nThe above bounds are quite recent and therefore the Ramsey-Tur\\'an number\\linebreak $\\textup{\\textbf{RT}}_s(n, H, f)$ was not extensively studied for small $f$. \nPreviously, most researchers investigated the Ramsey-Tur\\'an number for $f$ not too much smaller than $n$. Perhaps the first paper written about this problem was by S\\'os~\\cite{VS}. In particular there are many results on $\\theta_r(H)$, where\n\\[\n\\theta_r(H) = \\lim_{\\varepsilon \\rightarrow 0} \\lim_{n \\rightarrow \\infty} \\frac{1}{n^2}\\textup{\\textbf{RT}}(n, H, \\varepsilon n).\n\\] \nIt is perhaps surprising that $\\theta_r(H)$ would ever be positive, but it is known for example that $\\theta_2(K_4)= \\frac{1}{8}$ (upper bound by Szemer\\'edi in \\cite{Sz}, lower bound by Bollob\\'as and Erd\\H{o}s in \\cite{BE}). Several other exact values for $\\theta_r(H)$ are known, and even more bounds are known where exact values are not, see, for example, the papers by Balogh and Lenz~{\\cite{BL,BL2}}; Erd\\H{o}s, Hajnal, Simonovits, S\\'os, and Szemer\\'edi~\\cite{EHSSS}; \nErd\\H{o}s, Hajnal, S\\'os, and Szemer\\'edi~\\cite{EHSS}, and Simonovits and S\\'os~\\cite{SS}. Until recently only a very few results for $f \\le n^{\\delta}$ with $\\delta<1$ were given. For example, Sudakov~\\cite{SU} gave upper and lower bounds on $\\textup{\\textbf{RT}}_2(n, K_{4}, n^{\\delta})$ (see also \\cite{SU3} and~\\cite{FLZ}). Balogh, Hu, and Simonovits~\\cite{BHS} studied $\\textup{\\textbf{RT}}_2(n,K_t,f)$ for several pairs of $t$ and $f$.\n\n\n\nIn this paper we study $\\textup{\\textbf{RT}}_s(n, K_{s+r}, f)$ for $\\textup{\\textbf{f}}_{s,s+r}(n) +1 \\le f\\le n^{\\delta}$ with $\\delta<1$. In view of \\eqref{f:1} and \\eqref{f:r}, it is of interest to ask about $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ for $r\\ge 1$ and $1\/2 < \\delta < 1$. Moreover, for $r\\ge 2$ it also makes sense to ask for $\\textup{\\textbf{RT}}_s(n, K_{s+r}, Cn^{1\/2})$ for a sufficiently large constant $C$ (which may depend on $s$). We will show (Theorem~\\ref{thm:main1}, \\ref{thm:main2}, and \\ref{thm:ub_delta}) that for all $r\\ge 1$, $\\varepsilon>0$ and $1\/2 < \\delta < 1$, and all sufficiently large $s$,\n\\[\n\\Omega\\of{n^{2 - (1-\\delta)\/r - \\varepsilon}} = \\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta) <\n\\begin{cases}\nn^{2-\\frac{(1-\\delta)^2}{r-\\delta}}, & \\text{ if $\\frac{r-\\delta}{1-\\delta}$ is an integer},\\\\ \nn^{2-\\frac{(1-\\delta)^2}{r+1-2\\delta}}, & \\text{ otherwise}. \n\\end{cases}\n\\]\nIn particular, this implies for $r=1$ and $1\/2 < \\delta < 1$ nearly optimal bounds,\n\\[\n\\Omega\\of{n^{1+\\delta - \\varepsilon}} = \\textup{\\textbf{RT}}_s(n, K_{s+1}, n^\\delta) = O\\of{n^{1+\\delta}}.\n\\]\nAs a matter of fact the upper bound is trivial since as it was already observed in~\\cite{EHSSS} if $G$ is $K_{s+1}$-free, then $\\Delta(G) < \\alpha_s(G)$.\nWe will also show that for specific values of $\\delta$ one can remove $\\varepsilon$ from the exponent (see Theorem~\\ref{thm:quad} and Corollary~\\ref{cor:quad}) and we conjecture that this should be true for all $1\/2 < \\delta < 1$.\n\nVery recently Balogh, Hu, and Simonovits~\\cite{BHS} introduced an interesting concept of a (Ramsey-Tur\\'an) phase transition. Following this direction we will study behavior of $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, f)$. In particular, for $1\/2 < \\delta < 1$ we show that $\\textup{\\textbf{RT}}_s(n, K_{2s}, n^\\delta)$ is subquadratic, while $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, n^\\delta)$ is quadratic (and we actually find its value asymptotically exactly). But for $\\delta = 1\/2$, $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, n^\\delta)$ is subquadratic again, yielding a ``jump'' in the critical window.\n\nOur proofs of the lower bounds are built upon the ideas in~\\cite{DR,DRR,Wo} and are based on certain models of random graphs that are constructed using some finite geometries. Roughly speaking, finite geometries provide us with a structure that allows us to bound the number of vertices that interact in certain ways, which helps us show that in the random graph we construct, we do not expect to see the forbidden subgraph (say if we are doing $\\textup{\\textbf{RT}}_s(n, K_{s+r}, f)$ then the forbidden subgraph is $K_{s+r}$) while we do expect to see many copies of $K_s$. The proofs use the probabilistic method, but the use of probability is relatively elementary. The proofs of the upper bounds use the dependent random choice technique (see, e.g., \\cite{FS}).\n\nThe rest of the paper is structured as follows. In Section~\\ref{sec:prelim} we state the probabilistic tools we will use. In Sections~\\ref{sec:main1} and~\\ref{sec:main2} we prove lower bounds on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^{\\delta})$ for $\\delta$ close to $1\/2$ (in Section~\\ref{sec:main1}) and then for larger values $\\delta <1$ (in Section~\\ref{sec:main2}). In Section~\\ref{sec:ub} we prove upper bounds on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^{\\delta})$. In Section~\\ref{sec:r_eq_1} we further discuss the case $r=1$ and show that for a few values $\\delta$ we can prove upper and lower bounds matching up to a constant factor. In Section~\\ref{sec:big_r} we discuss a phase transition of $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, f)$.\n\n\\section{Preliminaries}\\label{sec:prelim}\n\nThis paper uses the probabilistic method in the most classical sense: if we define a random structure and show that with some positive probability the random structure has a certain property, then there must exist a structure with that property. The probabilistic aspect of this paper is elementary. We use only standard bounds on the probability of certain events, which we state here. \n\nWe state basic forms of the Chernoff and Markov bounds (see, e.g., \\cite{AS,JLR}).\n\\begin{c2}\nIf X is any nonnegative random variable and $\\zeta > 0$, then\n\\[\n\\Pr\\left(X \\geq \\zeta\\cdot \\mathbb{E}(X)\\right) \\leq \\frac{1}{\\zeta}.\n\\]\n\\end{c2}\n\nLet $\\textrm{Bin}(n,p)$ denotes the random variable with binomial distribution with number of trials~$n$ and probability of success~$p$.\n\\begin{c1}\nIf $X \\sim \\textrm{Bin}(n,p)$ and $0 < \\varepsilon \\leq \\frac{3}{2}$, then\n\\[\n\\Pr \\left( |X - \\mathbb{E}(X)| \\geq \\varepsilon \\cdot \\mathbb{E}(X) \\right) \\leq 2 \\exp \\left\\{ -\\frac{\\mathbb{E}(X) \\varepsilon^2 }{3} \\right\\} .\n\\]\n\\end{c1}\n\nWe will also use the union bound.\n\\begin{ub}\nIf $E_i$ are events, then \n\\[\n\\Pr \\Big( \\bigcup_{i=1}^k E_i \\Big) \\leq k \\cdot \\max \\{\\Pr(E_i): i \\in [k]\\}.\n\\]\n\\end{ub}\n\nFinally, we say that an event $E_n$ occurs \\emph{with high probability}, or {\\textit{w.h.p. }} for brevity, if $\\lim_{n\\rightarrow\\infty}\\Pr(E_n)=1$.\n\nAll logarithms in this paper are natural (base $e$). Asymptotic notation can be viewed as either in the variable $n$ (the number of vertices in the graphs we are interested in) or $q$ (another parameter that will go to infinity along with $n$). When we say that a statement \\emph{holds for $s$ sufficiently large}, we mean that there exists some $s_0$ (which may dependent on some other parameters) such that the statement holds for any $s \\ge s_0$. For simplicity, in the asymptotic notion we do not round numbers that are supposed to be integers either up or down. This is justified since these rounding errors are negligible.\n\n\n\\section{Lower bound on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ for small values of $\\delta$}\\label{sec:main1}\n\nIn this section we will construct (randomly) a graph $G_1$ that gives our lower bound for $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ for relatively small $\\delta$ (not much bigger than $1\/2$). The construction uses several ideas from~\\cite{DR,DRR,Wo}.\n\nWe start with the affine plane, which is a hypergraph with certain desirable properties. For this range of $\\delta$ we want a somewhat sparser hypergraph, so we randomly remove some edges from the affine plane to form a new hypergraph $\\mathcal{H}_1$. We then construct $G_1$ by taking the vertices in each edge of $\\mathcal{H}_1$ and putting a complete $s$-partite graph (with a random $s$-partition) on them together with a large independent set. Consequently, $G_1$ will have many copies of $K_s$. On the other hand, since each edge of $\\mathcal{H}_1$ contains only copies of $K_s$ (and no larger complete subgraph), any possible copy of $K_{s+r}$ in $G_1$ must not be entirely contained in one edge of $\\mathcal{H}_1$. We will exploit the properties $\\mathcal{H}_1$ and how its edges interact to show that this is unlikely, and therefore we do not expect to see any $K_{s+r}$ in $G_1$.\n\n\\subsection{The hypergraph $\\mathcal{H}_1$} \\label{sec:H1}\n\nThe {\\em affine plane} of order $q$ is an incidence structure on a set of $q^2$ points and a set of $q^2+q$ lines such that: any two points lie on a unique line; every line contains~$q$ points; and every point lies on~$q+1$ lines. It is well known that affine planes exist for all prime power orders. (For more details see, e.g., \\cite{CA}.)\nClearly, an incidence structure can be viewed as a hypergraph with points corresponding to vertices and lines corresponding to hyperedges; we will use this terminology interchangeably.\n\nIn the affine plane, call lines $L$ and $L'$ \\emph{parallel} if $L \\cap L' = \\emptyset$. In the affine plane there exist $q+1$ sets of $q$ pairwise parallel lines. Let $\\mathcal{H}=(V, \\mathcal{L})$ be the hypergraph obtained by removing a parallel class of $q$ lines from the affine plane or order~$q$. Thus, $\\mathcal{H}$ is $q$-regular hypergraph of order~$q^2$.\n\nThe objective of this section is to establish the existence of a certain hypergraph $\\mathcal{H}_1 = (V_1,\\mathcal{L}_1) \\subseteq \\mathcal{H}$ by considering a random sub-hypergraph of $\\mathcal{H}$. Preceding this, we introduce some terminology. \nCall $S \\subseteq V_1$ \\emph{complete} if every pair of points in $S$ is contained in some common line in~$\\mathcal{L}_1$. \nWe distinguish 2 types of complete \\emph{dangerous} subsets $S\\subseteq V$. \\emph{Type~1} dangerous set consists of $|S|$ points in general position. \\emph{Type~2} dangerous set consists of $|S|-r$ points that lie on some line $L\\in\\mathcal{L}_1$ and a set $R$ of $r$ many other points that do not belong to $L$. \n\n\n\n\n\\begin{lemma} \\label{lem:1}\nLet $q$ be a sufficiently large prime and $\\log q \\ll \\lambda \\le q$. \nThen, there exists a $q$-uniform hypergraph $\\mathcal{H}_1=(V_1,\\mathcal{L}_1)$ of order $q^2$ such that:\n\\begin{enumerate}[label=$({\\textup{H}}_{1}\\textup{\\alph*})$]\n\\item\\label{H1:a} Any two vertices are contained in at most one hyperedge;\n\\item\\label{H1:b} For every $v \\in V_1$, $\\frac{\\lambda}{2} \\le \\deg_{\\mathcal{H}_1}(v) \\leq \\frac{3\\lambda}{2}$; \n\\item\\label{H1:c} $|\\mathcal{D}_a| \\leq \\frac{4\\lambda^{a^2\/2} }{ q^{(a^2-5a)\/2}}$ and $| \\mathcal{D}_b| \\leq \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}} $, where $\\mathcal{D}_a$ is the set of all dangerous sets of Type 1 and size~$a$, and $\\mathcal{D}_b$ is the set of all dangerous sets of Type 2 and size~$b$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} Let $V_1 = V$ be the same vertex set as $\\mathcal{H}$, and let $\\mathbb{H}_1=(V_1,\\mathcal{L}_1)$ be a random sub-hypergraph of $\\mathcal{H}$ where every line in $\\mathcal{L}$ is taken independently with probability \n${\\lambda}\/{q}$.\n\nSince $\\mathbb{H}_1$ is a subgraph of $\\mathcal{H}$, any two vertices are in at most one line, so $\\mathbb{H}_1$ always satisfies \\ref{H1:a}. \nWe will show that $\\mathbb{H}_1$ satisfies {\\textit{w.h.p. }} \\ref{H1:b} and satisfies \\ref{H1:c} with probability at least $1\/2$. \nTogether this implies that $\\mathbb{H}_1$ satisfies \\ref{H1:a}-\\ref{H1:c} with probability at least $1-\\frac{1}{2}-o(1)$, establishing the existence of a hypergraph $\\mathcal{H}_1$ that satisfies \\ref{H1:a}-\\ref{H1:c}. \n\n\\medskip\n\n\\textbf{\\ref{H1:b}:} Observe for fixed $v \\in V_1$, $\\deg_{\\mathbb{H}_1}(v) \\sim \\textrm{Bin}(q, \\frac{\\lambda}{q})$ and the expected value $\\mathbb{E}(\\deg_{\\mathbb{H}_1}(v))= \\lambda.$ So by the Chernoff bound with $\\varepsilon = \\frac{1}{2}$, \n\\[\n \\Pr \\Big( |\\deg_{\\mathbb{H}_1}(v)-\\lambda| \\geq \\frac{\\lambda}{2} \\Big) \\leq 2 \\exp \\left\\{ -\\frac{\\lambda}{12} \\right\\}. \n\\]\nThus by the union bound the probability that there exists some $v \\in V_1$ with $\\deg_{\\mathbb{H}_1}(v) \\notin \\sqbs{\\frac{\\lambda}{2} , \\frac{3\\lambda}{2}}$ is at most\n\\[\nq^2 \\cdot 2 \\exp \\left\\{-\\frac{\\lambda}{12}\\right\\} = 2 \\exp \\left\\{2 \\log q-\\frac{\\lambda}{12}\\right\\} = o(1),\n\\]\nsince $\\lambda\\gg\\log q$.\n\n\\medskip\n\n\\textbf{\\ref{H1:c}:} In order to show we have both $|\\mathcal{D}_a| \\leq \\frac{4\\lambda^{a^2\/2} }{ q^{(a^2-5a)\/2}}$ and $| \\mathcal{D}_b| \\leq \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}} $ with probability at least $\\frac{1}{2}$, we begin by counting the number of dangerous subsets of each type. Clearly the number of Type 1 dangerous subsets is at most ${q^2 \\choose a}$ and each of them contains $\\binom{a}{2}$ lines. To count the number of Type 2 dangerous subsets first we choose a line $L$ out of $q^2$ lines in~$\\mathcal{L}$. Next we choose $b-r$ points in $L$ having $\\binom{q}{b-r}$ choices, and finally, we choose the remaining $r$ points not in $L$. Thus, the number of configurations of Type~2 is at most $\\binom{q^2}{1} \\cdot \\binom{q}{b-r} \\cdot q^{2r}$. Now we bound the number of lines in a dangerous set of Type 2. First there is the line $L$ containing $|S|-r$ points. Then, every pair of points $u,v$, where $u \\in L \\cap S$ and $v \\in R$, must be contained in some line $L_{u,v}$ in $\\mathcal{L}$. No line $L_{u,v}$ can contain more than one vertex in $L$, but it is possible for some line $L_{u,v}$ to contain multiple vertices in $R$. However for $v, v' \\in R$, if $L_{u,v}$ contains $v'$ then no other line can contain both $v, v'$. Thus, the number of lines of the form $L_{u,v}$ such that $|L_{u,v} \\cap R|\\ge 2$ is at most $\\binom{r}{2}$. Now since each of the $r(b-r)$ many pairs $\\{u,v\\}$ must be covered by some $L_{u,v}$, and the number of lines covering multiple pairs is at most $\\binom{r}{2}$, and no line covers more than $r$ many pairs, the total number of lines $L_{u,v}$ is at least $r(b-r) - r\\binom{r}{2} \\ge br - 2r^3$. \n\nBy the linearity of expectation, we now compute \n\\[\n\\mathbb{E}(| \\mathcal{D}_a|) \n\\leq \\binom{q^2}{a} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{\\binom{a}{2}} \\leq q^{2a} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{\\binom{a}{2}} \\le \\frac{\\lambda^{a^2 \/2 }}{ q^{(a^2-5a)\/2}}\n\\]\nand\n\\begin{align*}\n\\mathbb{E}(|\\mathcal{D}_b|) \n&\\leq \\binom{q^2}{1} \\cdot \\binom{q}{b-r} \\cdot q^{2r} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{br - 2r^3}\\\\\n&\\leq q^{2} \\cdot q^{b-r} \\cdot q^{2r} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{br - 2r^3} \\le \\frac{\\lambda^{br } }{ q^{b(r-1) - 5r^3}}.\n\\end{align*}\nThus, the Markov bound yields \n\\[\n\\Pr \\left( |\\mathcal{D}_a| \\ge \\frac{4\\lambda^{a^2 \/2 }}{ q^{(a^2-5a)\/2}} \\right)\n\\le \\Pr \\left( |\\mathcal{D}_a| \\ge 4 \\mathbb{E} ( |\\mathcal{D}_a|) \\right) \\le \\frac{1}{4}\n\\]\nand \n\\[\n\\Pr \\left( | \\mathcal{D}_b| \\ge \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}}\\right)\n\\le \\Pr \\left( |\\mathcal{D}_b| \\ge 4 \\mathbb{E} ( | \\mathcal{D}_b|) \\right) \\le \\frac{1}{4},\n\\]\nand finally\n\\[\n\\Pr \\left( |\\mathcal{D}_a| \\le \\frac{4\\lambda^{a^2 \/2 }}{ q^{(a^2-5a)\/2}} \\mbox{ and } | \\mathcal{D}_b| \\le \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}}\\right) \\ge 1 - \\frac 14 - \\frac 14 = \\frac 12,\n\\]\nas required.\n\\end{proof}\n\n\n\\subsection{The graph $G_1$}\\label{sec:G1}\nBased upon the hypergraph $\\mathcal{H}_1$ established in the previous section, we will construct a graph $G_1$ with the following properties. \n\\begin{lemma} \\label{lem:3}\nLet $r\\ge 1$ and $s$ be sufficiently large constant.\nLet $q$ be a sufficiently large prime, $q \\ge \\lambda \\gg \\log q$, $ 1 \\ge p \\gg (\\log q) \/ \\lambda$, and $\\alpha \\ge (10 s \\log s)q\/p$. Furthermore, let $a$ be a positive constant and $b=\\left\\lceil(s+r) \/{ \\binom{a-1}{2} }\\right\\rceil +r$. Then, there exists a graph $G_1$ of order $q^2$ such that:\n\\begin{enumerate}[label=$({\\textup{G}}_{1}\\textup{\\alph*})$]\n\\item\\label{G1:a} $\\alpha_s(G_1) < \\alpha$;\n\\item\\label{G1:b} For every vertex $v$ in $G_1$, $ \\deg_{G_1}(v)= \\Theta( \\lambda q p^2)$;\n\\item\\label{G1:c} $G_1$ has at most $8\\left(\\frac{\\lambda^{a^2 \/2} p^{a^2 -a}}{ q^{(a^2-5a)\/2}} + \\frac{\\lambda^{br } p^{(2r+1)b - 4r^3}}{ q^{b(r-1) -5r^3}}\\right)$ copies of $K_{s+r}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} Fix a hypergraph $\\mathcal{H}_1=(V_1,\\mathcal{L}_1)$ as established by Lemma~\\ref{lem:1}. Construct the random graph $\\mathbb{G}_1=(V_1,E)$ as follows. For every $L\\in \\mathcal{L}_1$, let $\\chi_L : L \\to {[s+1]}$ be a random partition of the vertices of $L$ into $ {s+1}$ classes, where for every $v \\in L$, \n\\[\n\\Pr(\\chi(v) = i) = \n\\begin{cases}\n{p}\/{s} & \\text{for } 1\\le i\\le s,\\\\\n1-p & \\text{for } i=s+1,\n\\end{cases}\n\\]\nand $\\chi(v)$ is assigned independently from other vertices.\n If $\\chi_L(v) =s+1$, then we say that $v$ is {\\em $L$-isolated}. Let $\\{x,y\\} \\in E$ if $\\{x,y\\} \\subseteq L$ for some $L \\in \\mathcal{L}_1$ and $\\chi_L(x), \\chi_L(y)$ are distinct and neither $x$ nor $y$ is $L$-isolated. Thus for every $L \\in \\mathcal{L}_1$, $\\mathbb{G}_1[L]$ consists of a set of isolated vertices (the $L$-isolated vertices) together with a complete $s$-partite graph with vertex partition $L = \\chi^{-1}_L (1) \\cup \\chi^{-1}_L (2) \\cup \\dots \\cup \\chi^{-1}_L (s)$ (where the classes need not have the same size and the unlikely event that a class is empty is permitted). Observe that not only are $\\mathbb{G}_1[L]$ and $\\mathbb{G}_1[L']$ edge disjoint for distinct $L, L' \\in \\mathcal{L}_1$, but also that the partitions for $L$ and $L'$ were determined independently.\n\n\nWe will show that $\\mathbb{G}_1$ satisfies {\\textit{w.h.p. }} \\ref{G1:a} and \\ref{G1:b} and satisfies \\ref{G1:c} with probability at least $1\/2$. \n\n\\medskip\n\n\\textbf{\\ref{G1:a}:} First fix $C \\in {V_1 \\choose \\alpha}$. We will bound the probability that $\\mathbb{G}_1[C] \\not \\supseteq K_s$. For a fixed $L$, the probability that one of the classes $\\chi_L^{-1}(1), \\dots , \\chi_L^{-1}(s)$ contains no element of $C$ is at most $s\\of{1-\\frac ps}^{|L \\cap C|} \\le s e^{-\\frac {p}{s} |L \\cap C|}$. Note that \n\\[\n\\sum_{L \\in \\mathcal{L}_1} |L \\cap C| \\ge \\frac 12 \\lambda |C| = \\frac 12 \\lambda \\alpha,\n\\] \nsince each point is in at least $\\frac 12 \\lambda$ lines due to condition~\\ref{H1:b}. Now since $\\chi_L$ and $\\chi_{L'}$ are chosen independently for $L \\neq L'$ we get, \n\\[\n\\Pr\\Big( K_s \\not \\subseteq \\mathbb{G}_1[C] \\Big) \\le \\prod_{L \\in \\mathcal{L}_1}\\Pr \\Big( K_s \\not \\subseteq \\mathbb{G}_1[L \\cap C] \\Big) \\le s^{|\\mathcal{L}_1|} \\exp\\left\\{ -\\frac ps \\sum_{L \\in \\mathcal{L}_1} |L \\cap C| \\right\\},\n\\]\nand since $|\\mathcal{L}_1| \\le 3\\lambda q$ by~\\ref{H1:b}, \n\\[\n\\Pr\\Big( K_s \\not \\subseteq \\mathbb{G}_1[C] \\Big) \\le \\exp \\left\\{(3\\log s) \\lambda q - \\frac{1}{2s} \\lambda \\alpha p \\right\\}.\n\\]\n\nSo by the union bound, the probability that there exists a set $C$ of $\\alpha$ vertices in $\\mathbb{G}_1$ that contains no $K_s$ is at most\n\\begin{align*}\n&q^{2 \\alpha} \\exp \\left\\{(3\\log s) \\lambda q - \\frac{1}{2s} \\lambda \\alpha p \\right\\}\\le \\exp \\left\\{2\\alpha \\log q +(3\\log s) \\lambda q - \\frac{1}{2s} \\lambda \\alpha p \\right\\}=o(1),\n\\end{align*}\nsince $ p \\gg (\\log q) \/ \\lambda$ and $\\alpha \\ge (10 s \\log s)q\/p$.\n\nThus, {\\textit{w.h.p. }} $\\alpha_s(\\mathbb{G}_1) < \\alpha$.\n\n\\medskip\n\n\\textbf{\\ref{G1:b}:} Observe that for a fixed $L\\in \\mathcal{L}_1$, the number of non-$L$-isolated vertices $|\\chi_L^{-1}([s])|$ is distributed as $\\textrm{Bin}(q, p)$ which has expectation $pq $ so by the Chernoff bound with $\\varepsilon = 1\/2$ get that\n\\[ \n\\Pr\\of{ \\left| |\\chi_L^{-1}([s])| -pq \\right| > \\frac 12 pq} \\le 2\\exp \\left\\{-\\frac{pq}{12} \\right\\}\n\\]\nand so by the union bound, the probability that there exists some $L$ such that\\linebreak $\\left| |\\chi_L^{-1}([s])| -pq \\right| > \\frac 12 pq$ is at most \n\\[\nq^2 \\cdot 2\\exp \\left\\{-\\frac{pq}{12} \\right\\} = 2\\exp \\left\\{2\\log q-\\frac{pq}{12} \\right\\} = o(1),\n\\] \nsince $p\\gg (\\log q) \/ \\lambda$ and $\\lambda \\le q$.\nThus, {\\textit{w.h.p. }} every line has between $\\frac 12 pq$ and $\\frac 32 pq$ many non-$L$-isolated vertices. \n\nNow for fixed $v$, let $X_v$ be the number of lines $L$ in which $v$ is non-$L$-isolated. $X_v$ is distributed as $\\textrm{Bin}(\\deg_{\\mathcal{H}_1} (v), p)$ which has expectation $\\deg_{\\mathcal{H}_1} (v)p \\ge \\lambda p\/2$. Now by the Chernoff bound with $\\varepsilon = 1\/2$ we get \n\\[\n\\Pr\\of{|X_v - \\deg_{\\mathcal{H}_1} (v)p| > \\frac 12 \\deg_{\\mathcal{H}_1} (v)p} \\le 2 \\exp \\left\\{-\\frac{\\deg_{\\mathcal{H}_1} (v)p}{12} \\right\\} \\le 2 \\exp \\left\\{-\\frac{\\lambda p}{24} \\right\\}\n\\]\nand so by the union bound, the probability that there exists some point $v$ with $|X_v - \\deg_{\\mathcal{H}_1} (v)p| > \\frac 12 \\deg_{\\mathcal{H}_1} (v)p$ is at most \n\\[\nq^2 \\cdot 2 \\exp \\left\\{-\\frac{\\lambda p}{24} \\right\\} = 2 \\exp \\left\\{2\\log q-\\frac{\\lambda p}{24} \\right\\} = o(1),\n\\]\nsince $p \\gg (\\log q) \/ \\lambda$.\nSo {\\textit{w.h.p. }} for every $v$, $v$ is non-$L$-isolated for some number of lines $L$ that is at least $\\frac 12 \\deg_{\\mathcal{H}_1} (v)p \\ge \\frac 14 \\lambda p$ and at most $\\frac 32 \\deg_{\\mathcal{H}_1} (v)p \\le \\frac 94 \\lambda p$.\n\nNow assume for each $L$ and each $v \\in L$ we have revealed whether $v$ is $L$-isolated, but we have not revealed $\\chi_L(v)$ when $v$ is non-$L$-isolated. When we do reveal values $\\chi_L(v)$ to form the graph $\\mathbb{G}_1$, we have for each non-$L$-isolated vertex $v$ that\n$\\deg_{\\mathbb{G}_1[L]}(v) \\sim \\textrm{Bin}(|\\chi_L^{-1}([s])|-1, \\frac{s-1}{s})$. Thus,\n$\\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v)) = (|\\chi_L^{-1}([s])|-1)(s-1)\/s \\ge (\\frac 12 pq-1)(s-1)\/s$ and the Chernoff bound with $\\varepsilon = 1\/2$ tells us that \n\\begin{align*}\n\\Pr\\of{|\\deg_{\\mathbb{G}_1[L]}(v) - \\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))| \\ge \\frac 12 \\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))} &\\le 2\\exp\\left\\{- \\frac{\\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))}{12} \\right\\}\\\\\n& \\le 2\\exp\\left\\{- \\frac{(\\frac 12 pq-1)(s-1)\/s}{12} \\right\\}\n\\end{align*}\nso by the union bound, the probability that there is a vertex $v$ with $|\\deg_{\\mathbb{G}_1[L]}(v) - \\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))| \\ge \\frac 12 \\mathbb{E}(\\deg_{\\mathbb{G}_1[L]}(v))$ is at most \n\\[\nq^2 \\cdot 2\\exp\\left\\{- \\frac{(\\frac 12 pq-1)(s-1)\/s}{12} \\right\\} = o(1),\n\\] \nsince $p \\gg (\\log q) \/ \\lambda$ and $\\lambda \\le q$. Thus {\\textit{w.h.p. }} for every vertex $v$ and line $L$ for which $v$ is non-$L$-isolated we have that $\\deg_{\\mathbb{G}_1[L]}(v)$ is at least $\\frac 12 (|\\chi_L^{-1}([s])|-1)(s-1)\/s \\ge \\frac 18 pq$ and at most $\\frac 32 (|\\chi_L^{-1}([s])|-1)(s-1)\/s \\le \\frac 94 pq$. \n\nThus, for each vertex $v$, {\\textit{w.h.p. }} its degree in $\\mathbb{G}_1$ is $\\Theta( \\lambda q p^2)$.\n\n\\medskip\n\n\\textbf{\\ref{G1:c}:} Recall that $a$ is some positive integer and $b=\\left\\lceil(s+r)\/{ \\binom{a-1}{2} }\\right\\rceil +r.$\nFirst we show that every copy of $K_{s+r}$ in $\\mathbb{G}_1$ contains a dangerous subset in $\\mathcal{D}_a \\cup \\mathcal{D}_b$.\nLet $K$ be any copy of $K_{s+r}$ in $\\mathbb{G}_1$. Clearly if $K$ contains a subset of $a$ points in general position then such subset is in $\\mathcal{D}_a$. Assume that $K$ contains at most $a-1$ points in general position. These points can be covered by at most $\\binom{a-1}{2}$ lines. In fact, all of the $s+r$ points must belong to those lines. Thus, there is a line $L$ with at least $\\lceil (s+r)\/\\binom{a-1}{2}\\rceil =b-r$ points from $K$. Moreover, since each line can contain at most $s$ points from $K$ there is a set $R$ of $r$ additional points in~$K$ that do not belong to~$L$. Hence, set $L\\cup R$ is in $\\mathcal{D}_b$. \n\nBy \\ref{H1:c}, $|\\mathcal{D}_a| \\leq \\frac{4\\lambda^{a^2 \/ 2} }{ q^{(a^2-5a)\/2}}$, and $| \\mathcal{D}_b| \\leq \\frac{4\\lambda^{br } }{ q^{b(r-1) - 5r^3}} $. We will show that {\\textit{w.h.p. }} none of these dangerous sets gives rise to a $K_{s+r}$ in $\\mathbb{G}_1$. In order for such dangerous set to give a $K_{s+r}$ in $\\mathbb{G}_1$, none of the vertices can be $L$-isolated in any of the lines in the dangerous set. For a Type 1 dangerous set, there are $\\binom{a}{2}$ lines, each containing $2$ vertices, so the probability that no vertex $v$ is $L$-isolated for $L$ containing $v$ is $p^{2 \\binom{a}{2}} = p^{a^2 - a}$. Thus, the expected number of copies of $K_{s+r}$ that arise from Type 1 dangerous sets is at most\n\\[\n\\frac{4\\lambda^{a^2\/2}}{ q^{(a^2-5a)\/2}}\\cdot p^{a^2-a}.\n\\]\nNow for dangerous sets of Type 2, the probability that a dangerous set in $\\mathcal{D}_b$ gives rise to a $K_{s+r}$ is $p^x$, where $x$ is the number of point-line incidences there are within the dangerous set. We observed before that each Type~2 dangerous set consists of a line~$L$ with $b-r$ points and a set $R$ of $r$~points and at least $r(b-r) - r\\binom{r}{2}$ lines containing one point from $L$ and at least one point from $R$. Thus, the number of point-line incidences is at least\n\\[\n(b-r) + 2 \\left(r(b-r) - r\\binom{r}{2} \\right) \\ge (2r+1)b - 4r^3.\n\\]\nTherefore, the expected number of copies of $K_{s+r}$ arising from Type 2 dangerous sets is at most\n\\[\n\\frac{4\\lambda^{br}}{ q^{b(r-1) - 5r^3}} \\cdot p^{(2r+1)b - 4r^3}.\n\\]\n\nThus by linearity of expectation the total number of copies of $K_{s+r}$ has expectation at most\n\\[\n\\frac{4\\lambda^{a^2\/2} p^{a^2-a}}{ q^{(a^2-5a)\/2}} + \\frac{4\\lambda^{br} p^{(2r+1)b - 4r^3}}{ q^{b(r-1) - 5r^3}}\n\\]\nand so we are done by the Markov bound applied with $\\zeta= 2$. \n\\end{proof}\n\n\n\n\n\\subsection{Deriving the lower bound for small values of $\\delta$} \\label{sec:LB1}\n\n\\begin{theorem}\\label{thm:main1}\nLet $r\\ge 1$, $\\varepsilon>0$, and $\\frac 12 < \\delta \\le \\frac 12 + \\frac{1}{2(2r+1)}$. Then for all sufficiently large $s$, we have \n\\begin{equation}\\label{thm:main1:eq1}\n\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta) = \\Omega\\of{n^{2 - (1-\\delta)\/r - \\varepsilon}}.\n\\end{equation}\nFurthermore, if $r\\ge 2$, then there exists a positive constant $C=C(s)$ such that for all sufficiently large $s$\n\\begin{equation}\\label{thm:main1:eq2}\n\\textup{\\textbf{RT}}_s(n, K_{s+r}, C\\sqrt{n}) = \\Omega\\of{n^{2 - 1\/(2r) - \\varepsilon}}.\n\\end{equation}\n\\end{theorem}\n\n\n\\begin{proof}\nWe will apply Lemma~\\ref{lem:3} after discussing how to set parameters. First we prove~\\eqref{thm:main1:eq1} assuming that $\\frac 12 < \\delta \\le \\frac 12 + \\frac{1}{2(2r+1)}$.\n\nFix $r\\ge 1$ and $\\varepsilon>0$. It is known that for large $x$ there exists a prime number between $x$ and $x(1+o(1))$ (see, e.g.,~\\cite{BHP}). Hence, for large $n$ there is a prime number $q$ such that $\\sqrt{n} \\le q \\le (1+o(1))\\sqrt{n}$. Set $\\alpha = n^\\delta$ and $p = \\kappa q^{-(2 \\delta -1)}$, where $\\kappa = 20s\\log s$. \nWe will show that the above parameters satisfy all assumptions of Lemma~\\ref{lem:3} implying the existence of a graph $G_1$ of order $q^2$ satisfying \\ref{G1:a}-\\ref{G1:c}.\n\nFirst observe that $(10 s \\log s)q\/p = q^{2\\delta} \/ 2 \\le n^{\\delta} = \\alpha$, as required by Lemma~\\ref{lem:3}. Thus, due to~\\ref{G1:a} the $s$-independence number of $G_1$ is less than $\\alpha$. \n\nSet $a =20r$, and $b=\\left\\lceil(s+r)\/ \\binom{a-1}{2} \\right\\rceil +r$. We will assume that $s$, and consequently $b $, is large enough such that for example\n\\begin{equation}\\label{thm:main1:eps}\n\\varepsilon > 5r^2\/b.\n\\end{equation}\nFurthermore, let\n\\[\n\\lambda = q^{1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b}.\n\\]\nObserve that \n\\[\n\\lambda p = \\kappa q^{1-1\/r + (2\\delta -1)(1+1\/r) - 10r^2\/b} \\gg \\log q,\n\\]\nsince $\\delta>1\/2$ and $b$ is sufficiently large. This implies that $p \\gg (\\log q) \/ \\lambda$, as required in Lemma~\\ref{lem:3}. Clearly this also implies that $\\lambda \\gg \\log q$. Also note that since $\\delta \\le \\frac 12 + \\frac{1}{2(2r+1)}$,\n\\[\n1- \\frac 1r + (2\\delta -1)\\of{2+ \\frac 1r} - \\frac{10r^2}{b} \\le 1 - \\frac 1r + \\frac{1}{2r+1}\\of{2+ \\frac 1r} - \\frac{10r^2}{b}=1 - \\frac{10r^2}{b}\n\\] \nso $\\lambda < q$ and hence all assumptions of Lemma~\\ref{lem:3} are satisfied.\n\nNow we show that $G_1$ is $K_{s+r}$-free. By~\\ref{G1:c} The number of copies of $K_{s+r}$ is at most \n\\begin{equation} \\label{nums+r}\n8 \\left(\\frac{\\lambda^{a^2 \/2} p^{a^2 -a}}{ q^{(a^2-5a)\/2}} + \\frac{\\lambda^{br } p^{(2r+1)b - 4r^3}}{ q^{b(r-1) -5r^3}}\\right).\n\\end{equation}\nThe order of magnitude of the first term in~\\eqref{nums+r} is \n\\begin{align*}\n\\frac{q^{\\left(1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b\\right) a^2\/2} \\cdot q^{(1-2\\delta)(a^2-a)}}{q^{(a^2-5a)\/2}}\n&= q^{(a^2\/2) \\cdot \\left( -(2-2\\delta)\/r - 10r^2\/b \\right) + (a\/2) \\cdot \\left( 2(2\\delta-1) +5\\right)} \\\\\n& \\le q^{-a^2\/(2r+1) + 7a\/2} = o(1),\n\\end{align*}\nwhere in the last line we used $\\delta \\le \\frac 12 + \\frac{1}{2(2r+1)}$ and $- 10r^2\/b < 0$, and the fact that $a=20r$. Now the second term of \\eqref{nums+r} has order of magnitude at most\n\\[\n\\frac{q^{\\left( 1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b\\right) \\cdot br} q^{(1-2\\delta)\\of{(2r+1)b-4r^3}} }{ q^{b(r-1) -5r^3}}\n=q^{-5r^3 + 4r^3(2\\delta-1)} \\le q^{-r^3} = o(1).\n\\]\n\nNow let $G$ be any induced subgraph of $G_1$ of order $n$. Clearly, $G$ is $K_{s+r}$-free with $\\alpha_s(G) < n^{\\delta}$. Furthermore, \n\\begin{align*}\n|E(G)| &\\ge |E(G_1)| - |V(G_1) - V(G)| \\cdot \\Delta(G_1)\\\\ \n& \\ge |V(G_1)|\\cdot \\delta(G_1) - o(1) \\cdot \\Delta(G_1) = \\Omega\\of{ q^2 \\cdot \\lambda q p^2},\n\\end{align*}\nby~\\ref{G1:b}. Finally observe that \n\\begin{align*}\nq^2 \\cdot \\lambda q p^2 = \\lambda q^3 p^2 &\\ge q^{1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b} \\cdot q^3 \\cdot \\of{ q^{1-2\\delta}}^2\\\\\n& \\ge q^{4-2(1-\\delta)\/r - 10r^2\/b} = n^{2-(1-\\delta)\/r - 5r^2\/b } > n^{2-(1-\\delta)\/r - \\varepsilon},\n\\end{align*}\nbecause of~\\eqref{thm:main1:eps}. Thus, $|E(G)| = \\Omega(n^{2-(1-\\delta)\/r - \\varepsilon})$ yielding the lower bound in~\\eqref{thm:main1:eq1}.\n\n\nThe proof of~\\eqref{thm:main1:eq2} is very similar. Assume that $r\\ge 2$ and let $\\delta=1\/2$, $p=1$, and $\\alpha = C \\sqrt{n}$ with $C=20s(\\log s)$. Other parameters are the same. Then, as in the previous case the assumptions of Lemma~\\ref{lem:3} hold.\n\\end{proof}\n\n\n\\section{Lower bound on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ for intermediate values of $\\delta$}\\label{sec:main2}\n\nRecall that in Section~\\ref{sec:main1} we started with the affine plane and made it sparser by taking edges with probability $\\lambda \/ q$. In Section~\\ref{sec:LB1}, to optimize our result we set \n\\[\n\\lambda = q^{1-1\/r + (2\\delta -1)(2+1\/r) - 10r^2\/b},\n\\]\nso when $\\delta = \\frac 12 + \\frac{1}{2(2r+1)}$, we are setting $\\lambda$ to be nearly $q$ (since $r^2 \/ b$ is small), which is about as large as $\\lambda$ can possibly be (and then we are making our hypergraph $\\mathcal{H}_1$ nearly as dense as the original affine plane). Thus it makes sense that if $\\delta$ is bigger than $\\frac 12 + \\frac{1}{2(2r+1)}$ then we no longer want a sparser version of the affine plane, but a denser version. Therefore in this section we will construct a denser hypergraph $\\mathcal{H}_2$ by keeping all of the edges and ``eliminating\" some vertices. That is the key difference between Sections~\\ref{sec:main1} and~\\ref{sec:main2}. Otherwise the proofs are quite similar.\n\n\\subsection{The hypergraph $\\mathcal{H}_2$} \\label{sec:H2}\n\nIn this section we establish the existence of a hypergraph $\\mathcal{H}_2$ with certain properties.\n\n\\begin{lemma} \\label{lem:4}\nLet $q$ be a sufficiently large prime and $\\log q \\ll \\lambda \\le q$. \nThen, there exists a $q$-regular hypergraph $\\mathcal{H}_2 = (V_2, \\mathcal{L}_2)$ of order $\\lambda q (1+o(1))$ such that:\n\\begin{enumerate}[label=$({\\textup{H}}_{2}\\textup{\\alph*})$]\n\\item\\label{H2:a} Any two vertices are contained in at most one hyperedge;\n\\item\\label{H2:b} For every $L \\in \\mathcal{L}_2$, $\\frac{\\lambda}{2} \\le |L| \\leq \\frac{3\\lambda}{2}$; \n\\item\\label{H2:c} $|\\mathcal{D}_a| \\leq 4\\lambda^a q^a$ and $| \\mathcal{D}_b| \\leq 4\\lambda^b q^{r+2}$, where $\\mathcal{D}_a$ is the set of all dangerous sets of Type~1 and size~$a$ and $\\mathcal{D}_b$ is the set of all dangerous sets of Type~2 and size~$b$.\n\\end{enumerate}\n\\end{lemma}\n\n\n\n\\begin{proof}\nStarting with the hypergraph $\\mathcal{H}$, we randomly ``eliminate\" some points. For each $v\\in V$ we randomly (and independently from other vertices) choose to eliminate $v$ with probability $1 - \\lambda \/ q$. Say $X$ is the set of vertices chosen for elimination. By ``elimination\", we mean that we will form a new hypergraph $\\mathbb{H}_2$ with vertex set $V_2 = V \\setminus X$, and edge set \n\\[\n\\mathcal{L}_2 = \\{L \\setminus X: L \\in \\mathcal{L} \\}.\n\\]\n\nFirst we will prove that {\\textit{w.h.p. }} $\\mathbb{H}_2$ has $\\lambda q (1+o(1))$ vertices. The number of vertices~$V$ is distributed as $\\textrm{Bin}(q^2, \\lambda \/ q)$ which has expectation $\\lambda q$. By the Chernoff bound with $\\varepsilon = (\\lambda q)^{-1\/4}$,\n\\[\n\\Pr\\of{|V - \\lambda q| > (\\lambda q)^{3\/4}} \\le 2 \\exp \\left\\{ - \\frac{(\\lambda q )^{1\/2}}{3}\\right\\} = o(1).\n\\]\n\nNow by the construction of $\\mathbb{H}_2$ and the properties of $\\mathcal{H}$, two vertices are in at most one line, so $\\mathbb{H}_2$ always satisfies \\ref{H2:a}. \nWe will show that $\\mathbb{H}_2$ satisfies {\\textit{w.h.p. }} \\ref{H2:b} and satisfies \\ref{H2:c} with probability at least $1\/2$. \nTogether this implies $\\mathbb{H}_2$ satisfies \\ref{H2:a}-\\ref{H2:c} with probability at least $1-\\frac{1}{2}-o(1)$, establishing the existence of a hypergraph $\\mathcal{H}_2$ that satisfies \\ref{H2:a}-\\ref{H2:c}. \n\n\\medskip\n\n\\textbf{\\ref{H2:b}:} Observe that for fixed $L \\in \\mathcal{L}_2$, $|L| \\sim \\textrm{Bin}(q, \\frac{\\lambda}{q})$ and $\\mathbb{E}(|L|)= \\lambda.$ So by the Chernoff bound with $\\varepsilon = \\frac{1}{2}$, \n\\[\n \\Pr \\Big( \\left||L|-\\lambda\\right| \\geq \\frac{\\lambda}{2} \\Big) \\leq 2 \\exp \\left\\{ -\\frac{\\lambda}{12} \\right\\}. \n\\]\nThus by the union bound the probability that there exists some $L \\in \\mathcal{L}_2$ with $|L| \\notin \\sqbs{\\frac{\\lambda}{2} , \\frac{3\\lambda}{2}}$ is at most\n\\[\nq^2 \\cdot 2 \\exp \\left\\{-\\frac{\\lambda}{12}\\right\\} = 2 \\exp \\left\\{2 \\log q-\\frac{\\lambda}{12}\\right\\} = o(1).\n\\]\n\n\\medskip\n\n\\textbf{\\ref{H2:c}:} In order to show that both $|\\mathcal{D}_a| \\leq 4\\lambda^{a} q^a$ and $|\\mathcal{D}_b| \\leq 4\\lambda^b q^{r+2}$ with probability at least $\\frac{1}{2}$, we recall from Section~\\ref{sec:H1} the number of dangerous subsets of each type. The number of Type~1 dangerous subsets is at most ${q^2 \\choose a}$, and the number of Type~2 dangerous subsets is at most $\\binom{q^2}{1} \\cdot \\binom{q}{b-r} \\cdot q^{2r}$. In order for $\\mathbb{H}_2$ to inherit a dangerous set, none of its vertices can be eliminated. \nBy the linearity of expectation, we now compute \n\\[\n\\mathbb{E}(| \\mathcal{D}_a|) \\leq \\binom{q^2}{a} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{a} \\leq \\lambda^a q^a\n\\]\nand\n\\[\n\\mathbb{E}(|\\mathcal{D}_b|) \n\\leq \\binom{q^2}{1} \\cdot \\binom{q}{b-r} \\cdot q^{2r} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{b}\n\\leq q^{2} \\cdot q^{b-r} \\cdot q^{2r} \\cdot \\left( \\frac{\\lambda}{q} \\right)^{b} = \\lambda^b q^{r+2},\n\\]\nand we are done by the Markov bound applied twice with $\\zeta=4$.\n\\end{proof}\n\n\\subsection{The graph $G_2$}\\label{sec:G2}\n\nBased upon the hypergraph $\\mathcal{H}_2$ established in the previous section, we will construct a graph $G_2$ with the following properties. \n\n\\begin{lemma} \\label{lem:5}\nLet $r\\ge 1$ and $s$ be sufficiently large constant.\nLet $q$ be a sufficiently large prime, $q \\ge \\lambda \\gg \\log q $, $1 \\ge p \\gg (\\log q) \/ \\lambda$, and $\\alpha \\ge (10 s \\log s)q\/p$. Furthermore, let $a$ be a positive constant and $b=\\left\\lceil(s+r)\/\\binom{a-1}{2}\\right\\rceil +r$. Then, there exists a graph $G_2$ with $\\lambda q (1+o(1))$ vertices such that:\n\\begin{enumerate}[label=$({\\textup{G}}_{2}\\textup{\\alph*})$]\n\\item\\label{G2:a} $\\alpha_s(G_2) < \\alpha$;\n\\item\\label{G2:b} For every vertex $v$ in $G_2$, $ \\deg_{G_2}(v)= \\Theta( \\lambda q p^2)$;\n\\item\\label{G2:c} $G_2$ has at most $8\\left(\\lambda^a q^a p^{a^2 -a} + \\lambda^b q^{r+2} p^{(2r+1)b - 4r^3}\\right)$ copies of $K_{s+r}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nStarting with the hypergraph $\\mathcal{H}_2=(V_2,\\mathcal{L}_2)$, we form the random graph $\\mathbb{G}_2=(V_2,E)$ as follows. For every $L\\in \\mathcal{L}_2$, let $\\chi_L : L \\to {[s+1]}$ be a random partition of the vertices of $L$ into $ {s+1}$ classes, where for every $v \\in L$,\n\\[\n\\Pr(\\chi(v) = i) = \n\\begin{cases}\n{p}\/{s} & \\text{for } 1\\le i\\le s,\\\\\n1-p & \\text{for } i=s+1,\n\\end{cases}\n\\]\nand $\\chi(v)$ is assigned independently from other vertices.\nIf $\\chi_L(v) =s+1$, then we say that $v$ is {\\em $L$-isolated}. Let $\\{x,y\\} \\in E$ if $\\{x,y\\} \\subseteq L$ for some $L \\in \\mathcal{L}_2$ and $\\chi_L(x), \\chi_L(y)$ are distinct and neither $x$ nor $y$ is $L$-isolated. Thus for every $L \\in \\mathcal{L}_2$, $\\mathbb{G}_2[L]$ consists of a set of isolated vertices (the $L$-isolated vertices) together with a complete $s$-partite graph with vertex partition $L = \\chi^{-1}_L (1) \\cup \\chi^{-1}_L (2) \\cup \\dots \\cup \\chi^{-1}_L (s)$.\n\nWe will show that $\\mathbb{G}_2$ satisfies {\\textit{w.h.p. }} \\ref{G2:a} and \\ref{G2:b} and satisfies \\ref{G2:c} with probability at least $1\/2$. \n\n\\medskip\n\n\\textbf{\\ref{G2:a}:} First fix $C \\in {V_2 \\choose \\alpha}$. We will bound the probability that $\\mathbb{G}_2[C] \\not \\supseteq K_s$. For a fixed $L$, the probability that one of the classes $\\chi_L^{-1}(1), \\dots, \\chi_L^{-1}(s)$ contains no element of $C$ is at most $s\\of{1-\\frac ps}^{|L \\cap C|} \\le s e^{-\\frac {p}{s} |L \\cap C|}$. Note that \n\\[\n\\sum_{L \\in \\mathcal{L}_2} |L \\cap C| = q |C| = \\alpha q,\n\\] \nsince each point is in $q$ lines. Now since $\\chi_L$ and $\\chi_{L'}$ are chosen independently for $L \\neq L'$ we get, \n\\begin{align*}\n\\Pr\\Big( K_s \\not \\subseteq \\mathbb{G}_2[C] \\Big) &\\le \\prod_{L \\in \\mathcal{L}_2}\\Pr \\Big( K_s \\not \\subseteq \\mathbb{G}_2[L \\cap C] \\Big)\\\\\n& \\le s^{q^2} \\exp\\left\\{ -\\frac ps \\sum_{L \\in \\mathcal{L}_2} |L \\cap C| \\right\\} \\le \\exp \\left\\{(\\log s) q^2 - \\frac{1}{s} \\alpha q p \\right\\}\n\\end{align*} \n\nSo by the union bound, the probability that there exists a set $C$ of $\\alpha$ vertices in $\\mathbb{G}_2$ that contains no $K_s$ is at most\n\\[\nq^{2 \\alpha} \\cdot \\exp \\left\\{(\\log s) q^2 - \\frac{1}{s} \\alpha q p \\right\\} \\le \\exp \\left\\{2\\alpha \\log q +(\\log s) q^2 - \\frac{1}{s} \\alpha q p \\right\\}=o(1),\n\\]\nsince $p \\gg (\\log q) \/ \\lambda$, $\\lambda \\le q$, and $\\alpha \\ge (10 s \\log s) q\/p$. \n\nThus, {\\textit{w.h.p. }} $\\alpha_s(\\mathbb{G}_2) < \\alpha$.\n\n\\medskip\n\n\n\n\\textbf{\\ref{G2:b}:} Observe that for a fixed $L\\in \\mathcal{L}_2$, the number of non-$L$-isolated vertices $|\\chi_L^{-1}([s])|$ is distributed as $\\textrm{Bin}(|L|, p)$ which has expectation $p|L| $ so by the Chernoff bound with $\\varepsilon = 1\/2$ we get that\n\\[\n\\Pr\\of{ \\left| |\\chi_L^{-1}([s])| -p|L| \\right| > \\frac 12 p|L|} \\le 2\\exp \\left\\{-\\frac{p|L|}{12} \\right\\}\\le 2\\exp \\left\\{-\\frac{p\\lambda}{24} \\right\\}\n\\]\nand so by the union bound, the probability that there exists some $L$ such that\\linebreak \n$\\left| |\\chi_L^{-1}([s])| -p|L| \\right| > \\frac 12 p |L|$ is at most \n\\[\nq^2 \\cdot 2\\exp \\left\\{-\\frac{p \\lambda}{24} \\right\\} = o(1),\n\\]\nsince $p\\gg (\\log q) \/ \\lambda$.\nThus, {\\textit{w.h.p. }} every line $L$ has at least $\\frac 12 p|L| \\ge \\frac 14 p \\lambda$ and at most $\\frac 32 p|L| \\le \\frac 94 p \\lambda$ many non-$L$-isolated vertices. \n\nNow for fixed $v$, let $X_v$ be the number of lines $L$ in which $v$ is non-$L$-isolated. $X_v$ is distributed as $\\textrm{Bin}(q , p)$ which has expectation $qp $. Now by the Chernoff bound with $\\varepsilon = 1\/2$ we get \n\\[\n\\Pr\\of{|X_v - qp| > \\frac 12 qp} \\le 2 \\exp \\left\\{-\\frac{qp}{12} \\right\\}\n\\] \nand so by the union bound, the probability that there exists some point $v$ with $|X_v - qp| > \\frac 12 qp$ is at most \n\\[\nq^2 \\cdot 2 \\exp \\left\\{-\\frac{q p}{12} \\right\\} = o(1),\n\\]\nsince $ p \\gg (\\log q) \/ \\lambda$ and $\\lambda \\le q$.\nSo {\\textit{w.h.p. }} for every $v$, $v$ is non-$L$-isolated for some number of lines $L$ that is between $\\frac 12 qp $ and at most $\\frac 32qp$.\n\nNow assume for each $L$ and each $v \\in L$ we have revealed whether $v$ is $L$-isolated, but we have not revealed $\\chi_L(v)$ when $v$ is non-$L$-isolated. When we do reveal values $\\chi_L(v)$ to form the graph $\\mathbb{G}_2$, we have for each non-$L$-isolated vertex $v$ that\n$\\deg_{\\mathbb{G}_2[L]}(v) \\sim \\textrm{Bin}(|\\chi_L^{-1}([s])|-1, \\frac{s-1}{s})$. Thus,\n$\\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v)) = (|\\chi_L^{-1}([s])|-1)(s-1)\/s \\ge (\\frac 14 p\\lambda-1)(s-1)\/s$ and the Chernoff bound with $\\varepsilon = 1\/2$ tells us that \n\\begin{align*}\n\\Pr\\of{|\\deg_{\\mathbb{G}_2[L]}(v) - \\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))| \\ge \\frac 12 \\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))} &\\le 2\\exp\\left\\{- \\frac{\\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))}{12} \\right\\}\\\\ \n&\\le 2\\exp\\left\\{- \\frac{(\\frac 14 p\\lambda-1)(s-1)\/s}{12} \\right\\}\n\\end{align*}\nso by the union bound, the probability that there is a vertex $v$ with $|\\deg_{\\mathbb{G}_2[L]}(v) - \\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))| \\ge \\frac 12 \\mathbb{E}(\\deg_{\\mathbb{G}_2[L]}(v))$ is at most \n\\[\nq^2 \\cdot 2\\exp\\left\\{- \\frac{(\\frac 14 p\\lambda-1)(s-1)\/s}{12} \\right\\} = o(1),\n\\]\nsince $p \\gg (\\log q) \/ \\lambda$. Thus {\\textit{w.h.p. }} for every vertex $v$ and line $L$ for which $v$ is non-$L$-isolated we have that $\\deg_{\\mathbb{G}_2[L]}(v)=\\Theta(p \\lambda)$. \n\nThus, for each vertex $v$, {\\textit{w.h.p. }} its degree in $\\mathbb{G}_2$ is $\\Theta( \\lambda q p^2)$.\n\n\\medskip\n\n\\textbf{\\ref{G2:c}:} Recall that $a$ is some positive integer and $b=\\left \\lceil(s+r)\/ { \\binom{a-1}{2} }\\right \\rceil +r$. \nFirst we show that every $K_{s+r}$ in $\\mathbb{G}_2$ contains a dangerous subset in $\\mathcal{D}_a \\cup \\mathcal{D}_b$.\nLet $K$ be any copy of $K_{s+r}$ in $\\mathbb{G}_2$. Similarly to the graph $\\mathbb{G}_1$, we can conclude that the vertices of $K$ must contain a dangerous set in $\\mathcal{D}_a$ or in $\\mathcal{D}_b$.\n\nIn order for such a dangerous set to give a $K_{s+r}$ in $\\mathbb{G}_2$ none of the vertices can be $L$-isolated in any of the lines in the dangerous set. For a Type 1 dangerous set, there are $\\binom{a}{2}$ lines, each containing $2$ vertices, so the probability that no vertex $v$ is $L$-isolated for $L$ containing $v$ is $p^{2 \\binom{a}{2}} = p^{a^2 - a}$. Thus, the expected number of copies of $K_{s+r}$ that arise from Type 1 dangerous sets is at most $4\\lambda^{a}q^a p^{a^2-a}.$ \nNow for dangerous sets of Type 2, the probability that a dangerous set in $\\mathcal{D}_b$ gives rise to a $K_{s+r}$ is $p^x$, where $x$ is the number of point-line incidences there are within the dangerous set. As in Section~\\ref{sec:G1} the number of point-line incidences is at least $ (2r+1)b - 4r^3$. Therefore the expected number of copies of $K_{s+r}$ arising from Type~2 dangerous sets is at most $4\\lambda^b q^{r+2} p^{(2r+1)b - 4r^3}$.\n\nThus by linearity of expectation the total number of copies of $K_{s+r}$ has expectation at most \n\\[\n4\\lambda^{a}q^a p^{a^2-a} + 4\\lambda^b q^{r+2} p^{(2r+1)b - 4r^3}\n\\] \nand so we are done by the Markov bound applied with $\\zeta=2$. \n\n\n\\end{proof}\n\n\\subsection{Deriving the lower bound for intermediate values of $\\delta$}\n\n\\begin{theorem}\\label{thm:main2}\nLet $r\\ge 1$, $\\varepsilon>0$, and $\\frac 12 + \\frac{1}{2(2r+1)} < \\delta < 1$. Then for all sufficiently large $s$, we have \n\\[\n\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta) = \\Omega\\of{n^{2 - (1-\\delta)\/r - \\varepsilon}}.\n\\]\n\\end{theorem}\n\nFirst we state a proposition that we will use to estimate fractions that have ``error\" in the numerator and denominator. \n\n\\begin{proposition} \\label{estlem}\n\nFor any real numbers $x, y, \\epsilon_x, \\epsilon_y$, if $x,y \\neq 0$ and $|\\frac{\\epsilon_x}{x}|, |\\frac{\\epsilon_y}{y}| \n\\le \\frac{1}{2}$, then \n\\[\n\\left|\\frac{x+ \\epsilon_x}{y+\\epsilon_y} - \\frac{x}{y} \\right| \\le \\frac{|\\epsilon_x y| + 3|\\epsilon_y x|}{y^2}. \n\\]\n\\end {proposition}\n\n\n\\begin{proof}\nObserve that \n\\begin{align*}\n\\left|\\frac{x+ \\epsilon_x}{y+\\epsilon_y} - \\frac{x}{y}\\right| &= \\left|\\frac{x}{y} \\left[ \\left(1+\\frac{\\epsilon_x}{x} \\right) \\cdot \\frac{1}{1+\\frac{\\epsilon_y}{y}} -1 \\right] \\right|\\\\\n&= \\left|\\frac{x}{y} \\left[ \\left(1+\\frac{\\epsilon_x}{x} \\right) \\cdot \\left(1 + \\sum_{n=1}^{\\infty} \\of{-\\frac{\\epsilon_y}{y}}^n \\right) -1 \\right]\\right|\\\\\n&= \\left|\\frac{x}{y} \\left[ \\frac{\\epsilon_x}{x} +\\of{1+\\frac{\\epsilon_x}{x}}\\sum_{n=1}^{\\infty} \\of{-\\frac{\\epsilon_y}{y}}^n \\right]\\right|\\\\\n&\\le\\left| \\frac{ \\epsilon_x }{y}\\right| + \\left| \\frac{x}{y} \\right| \\cdot \\left|1+\\frac{\\epsilon_x}{x}\\right| \\cdot \\sum_{n=1}^{\\infty} \\left|\\frac{\\epsilon_y}{y}\\right|^n\\\\\n&\\le\\left| \\frac{ \\epsilon_x }{y}\\right| + \\left| \\frac{x}{y} \\right| \\cdot \\frac{3}{2} \\cdot 2 \\left|\\frac{\\epsilon_y}{y}\\right|.\n\\end{align*}\nIn the last line we have used $|\\epsilon_x \/ x| \\le 1\/2$ and\n\\[\n\\sum_{n=1}^{\\infty} \\left|\\frac{\\epsilon_y}{y}\\right|^n = \\left|\\frac{\\epsilon_y}{y}\\right| \\sum_{n=0}^{\\infty} \\left|\\frac{\\epsilon_y}{y}\\right|^n \\le \\left|\\frac{\\epsilon_y}{y}\\right| \\cdot 2\n\\] \nwhich follows from $|\\epsilon_y \/ y| \\le 1\/2$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main2}]\nWe will apply Lemma~\\ref{lem:5} after discussing how to set parameters.\n\nFix $r\\ge 1$, $\\varepsilon>0$ and $\\frac 12 + \\frac{1}{2(2r+1)} < \\delta < 1$. Set \n\\[\na = 2 + \\max\\left\\{ \\left\\lceil \\frac{1}{\\delta} \\right\\rceil , \\;\\;\\left\\lceil \\of{\\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + 1} \\cdot \\frac{\\delta(2r+1)-1}{1-\\delta} \\right\\rceil \\right\\}\n\\]\nand\n\\[\nb=\\left \\lceil(s+r) \/ \\binom{a-1}{2} \\right \\rceil +r.\n\\] \nWe will assume that $s$, and consequently $b$, is large enough which satisfies for instance the following\n\\begin{equation}\\label{thm:main2:eps}\n\\varepsilon > 104r^2\/b.\n\\end{equation}\n\nSimilarly as in the proof of Theorem~\\ref{thm:main1} for large $n$ there is a prime number $q$ such that \n\\[\nn^{1\/\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}} \\le q\n\\le (1+o(1)) n^{1\/\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}.\n\\]\n(It will be shown soon that the exponents are positive.)\nLet \n\\[\n\\lambda = q^\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} \\quad \\text{ and }\\quad \np = \\kappa q^{1 - \\delta \\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}},\n\\]\nwhere $\\kappa = 20 s \\log s$. Finally set $\\alpha = n^\\delta $. \n\nWe will show that the above parameters satisfy all assumptions of Lemma~\\ref{lem:5} implying the existence of a graph $G_2$ of order $\\lambda q (1+o(1))$ satisfying \\ref{G2:a}-\\ref{G2:c}.\n\nFirst observe that \n\\[\n\\frac{(10s\\log s) q}{p} = \\frac{q^{\\delta\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}}{2}\n=\\frac{(1+o(1))n^{\\delta}}{2} \\le \\alpha.\n\\]\nyielding by~\\ref{G2:a} that the $s$-independence number of $G_2$ is less than $\\alpha$. \n\nNow we examine the exponent of~$\\lambda$. Clearly, \n\\[\n\\lim_{b\\to\\infty} \\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} = \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1}.\n\\]\nWe will show that for $b \\ge 20r^2$ the exponent is very close to the above limit. Indeed, since $r \\ge 1$ and $1\/2 < \\delta < 1$, we have \n\\[\n\\left|\\frac{ -4(1-\\delta)r^3 +r+3}{(1-\\delta)(2r+1)b} \\right| \\le \\frac{8r^3}{rb} \\le \\frac 12 \\quad \\text{ and } \\quad \\left|\\frac{-4\\delta r^3}{(\\delta(2r+1)-1)b} \\right| \\le \\frac{8r^3}{rb} \\le \\frac 12\n\\]\nand so by Proposition~\\ref{estlem} we obtain\n\\begin{align*}\n&\\left|\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{(\\delta(2r+1)-1)b - 4\\delta r^3} - \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} \\right| \\\\ \n&\\qquad\\qquad\\qquad\\le \\frac{\\of{4(1-\\delta)r^3 +r+3}(\\delta(2r+1)-1)b + 3\\cdot 4\\delta r^3(1-\\delta)(2r+1)b }{\\of{(\\delta(2r+1)-1)b}^2}\\\\\n&\\qquad\\qquad\\qquad\\le \\frac{\\of{8r^3}(2r)b + 12r^3(3r)b }{\\of{rb}^2} = \\frac{52r^2}{b},\n\\end{align*}\nwhere the last inequality follows from $r \\ge 1$ and $1\/2 < \\delta < 1$. Also note that \n\\[\n\\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} \\ge 1-\\delta\n\\] \nand\n\\[\n\\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} = 1- \\frac{(2\\delta-1)(2r+1)-1}{\\delta(2r+1)-1} \\le 1- \\frac{(2\\delta-1)(2r+1)-1}{2r}.\n\\]\nObserve that since $\\delta > \\frac 12 + \\frac{1}{2(2r+1)}$, we get $\\frac{(2\\delta-1)(2r+1)-1}{2r} > 0$. Thus, if \n\\[\n\\frac{52r^2}{b} < \\min\\left\\{1-\\delta, \\frac{(2\\delta-1)(2r+1)-1}{2r} \\right\\},\n\\] \nthen\n\\begin{equation}\\label{eq:main2:1}\n\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{(\\delta(2r+1)-1)b - 4\\delta r^3} \\le \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} + \\frac{52r^2}{b} < 1\n\\end{equation}\nand\n\\begin{equation}\\label{eq:main2:2}\n\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{(\\delta(2r+1)-1)b - 4\\delta r^3} \\ge \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} - \\frac{52r^2}{b} > 0.\n\\end{equation}\nConsequently, $\\lambda$ is at least $q^{\\Omega(1)}$ and less than $q$, so $\\log q \\ll \\lambda \\le q$, as required in Lemma~\\ref{lem:5}.\n\nFurthermore, \n\\[\n\\lambda p = \\kappa q^{1+(1-\\delta)\\of{ \\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} } - \\delta}\n\\ge \\kappa q^{1-\\delta} \\gg \\log q\n\\]\nand\n\\begin{align*}\np &= \\kappa q^{1 - \\delta \\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}\n\\overset{\\eqref{eq:main2:2}}{\\le} \\kappa q^{ 1-\\delta\\of{ \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} - \\frac{52r^2}{b} +1 } } \\\\\n&= \\kappa q^{ 1-\\delta\\of{ \\frac{1-\\delta}{\\delta} + \\frac{1-\\delta}{\\delta\\of{\\delta(2r+1)-1}} - \\frac{52r^2}{b} +1 } } \n= \\kappa q^{-\\frac{1-\\delta}{\\delta(2r+1)-1} + \\frac{52r^2\\delta}{b}} = o(1)\n\\end{align*}\nfor $b$ sufficiently large. Consequently, $1 \\ge p \\gg (\\log q) \/ \\lambda$ and all assumptions of Lemma~\\ref{lem:5} are satisfied.\n\n\nNow we will see that our choice of parameters makes $G_2$ a $K_{s+r}$-free graph. By~\\ref{G2:c}, the number of copies of $K_{s+r}$ is at most \n\\begin{equation} \\label{nums+r2}\n8\\left(\\lambda^a q^a p^{a^2 -a} + \\lambda^b q^{r+2} p^{(2r+1)b - 4r^3}\\right).\n\\end{equation}\nThe first term is on the order of $\\lambda^a q^a p^{a^2 -a} = O\\of{\\left(\\lambda q p^{a-1}\\right)^a}$.\nWe show that $\\lambda q p^{a-1} = o(1)$. The order of magnitude of the latter is\n\\begin{align*}\nq^\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} &\\cdot q \\cdot q^{\\of{1 - \\delta \\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}(a-1)}\\\\\n&=q^{\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} }\\cdot (1-\\delta(a-1)) + 1+(1-\\delta)(a-1)} \\\\\n&\\overset{\\eqref{eq:main2:1}}{\\le} q^{\\of{ \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + \\frac{52r^2}{b} }\\cdot (1-\\delta(a-1)) + 1+(1-\\delta)(a-1)} \\\\\n&=q^{ \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + \\frac{52r^2}{b}(1- \\delta(a-1)) + 1 + \\of{1 - \\delta - \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} \\delta }(a-1)} \\\\\n&\\le q^{ \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + 1 + \\of{1 - \\frac{(2r+1)\\delta}{\\delta(2r+1)-1} }(1-\\delta)(a-1)} \\\\\n&= q^{ \\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + 1 - \\frac{1}{\\delta(2r+1)-1} (1-\\delta)(a-1)} = o(1),\n\\end{align*}\nwhere the second to last line follows from $a > 1+ 1\/\\delta$ and the last line follows from\n\\[\na > 1 + \\of{\\frac{(1-\\delta)(2r+1)}{\\delta(2r+1)-1} + 1} \\cdot \\frac{\\delta(2r+1)-1}{1-\\delta}.\n\\] \nThus, $\\lambda^a q^a p^{a^2 -a}=o(1)$.\nNow we bound the order of the magnitude of the second term in~\\eqref{nums+r2}. Observe that \n\\begin{align*}\nq^{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3} b} &\\cdot q^{r+2} \\cdot q^{\\of{1 - \\delta \\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}}\\of{(2r+1)b - 4r^3}} = \\frac{1}{q} = o(1).\n\\end{align*}\nThus, $G_2$ is $K_{s+r}$-free.\n\nNow let $G$ be any induced subgraph of $G_2$ of order $n$. Clearly, $G$ is $K_{s+r}$-free with $\\alpha_s(G) < n^{\\delta}$. Furthermore, since $n = (1+o(1))\\lambda q$,\n\\begin{align*}\n|E(G)| &\\ge |E(G_2)| - |V(G_2) - V(G)| \\cdot \\Delta(G_2)\\\\ \n& \\ge |V(G_2)|\\cdot \\delta(G_2) - o(1) \\cdot \\Delta(G_2) = \\Omega\\of{ \\lambda q \\cdot \\lambda q p^2} = \\Omega(np^2),\n\\end{align*}\nby~\\ref{G2:b}. Since \n\\[\np = (1+o(1))\\kappa n^{1\/\\of{\\frac{(1-\\delta)(2r+1)b - 4(1-\\delta)r^3 +r+3}{\\left(\\delta(2r+1)-1\\right)b - 4\\delta r^3}+1}-\\delta}\n\\overset{\\eqref{eq:main2:1}}{\\ge} (1+o(1))\\kappa n^{1\/\\of{\\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} + \\frac{52r^2}{b}+1}-\\delta},\n\\]\nwe get\n\\[\n\\Omega\\of{n^2 p^2} \n= \\Omega \\of{n^{2-2\\delta + 2\/\\of{1+ \\frac{(1-\\delta)(2r+1) }{\\delta(2r+1)-1} + \\frac{52r^2}{b}}} }\n= \\Omega \\of{n^{2-2\\delta + 2\/\\of{\\frac{2r }{\\delta(2r+1)-1} + \\frac{52r^2}{b}}} }.\n\\]\nSince for any positive real numbers $x, y, z$ with $y > z$,\n\\[\n\\frac{x}{y+z} = \\frac{x}{y} \\cdot \\frac{1}{1+\\frac{z}{y}} \\ge \\frac xy \\cdot \\of{1- \\frac zy} = \\frac{x}{y} - \\frac{xz}{y^2},\n\\]\nwe get\n\\[\n2\\left\/\\right.\\!\\of{\\frac{2r }{\\delta(2r+1)-1} + \\frac{52r^2}{b}} \\ge \\frac{\\delta(2r+1)-1}{r} - \\frac{104r^2}{b} \\cdot \\of{\\frac{\\delta(2r+1)-1}{2r}}^2\n\\ge \\frac{\\delta(2r+1)-1}{r} - \\varepsilon,\n\\]\nthe latter is due to~\\eqref{thm:main2:eps}.\nThus,\n\\[\n\\Omega\\of{n^2 p^2} = \\Omega\\of{ n^{2-2\\delta + \\frac{\\delta(2r+1)-1}{r} - \\varepsilon} } = \\Omega\\of{n^{2-\\frac{1-\\delta}{r}-\\varepsilon}}\n\\]\ncompleting the proof.\n\\end{proof}\n\n\n\\section{Upper bound on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$}\\label{sec:ub}\n\nFirst we state the well-known dependent random choice lemma from a survey paper of Fox and Sudakov~\\cite{FS}. Early versions of this lemma were proved and applied by various researchers,\nstarting with Gowers~\\cite{Gowers}, R\\\"odl and Kostochka~\\cite{KRodl}, and Sudakov~\\cite{SU3,SU,SU2}.\n\n\\begin{lemma}\\label{lem:dependent}\nLet $a, d, m, n, r$ be positive integers. Let $G = (V,E)$ be a graph with $|V| = n$ vertices and\naverage degree $d = 2|E(G)|\/n$. If there is a positive integer $t$ such that\n\\[\n\\frac{d^t}{n^{t-1}} - n^r \\left( \\frac{m}{n} \\right)^{t} \\ge a,\n\\]\nthen $G$ contains a subset $U$ of at least $a$ vertices such that every $r$ vertices in $U$ have at least $m$\ncommon neighbors.\n\\end{lemma}\n\n\\begin{corollary}\\label{cor:sudakov}\nLet $r\\ge 1$ be an integer and $0<\\delta<1$.\nLet $G$ be a graph on $n$ vertices with at least $n^{2-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}$ edges. \nThen $G$ contains a subset $U$ of at least $n^{\\delta}$ vertices such that every $r$ vertices in $U$ have at least $n^{\\delta}$\ncommon neighbors.\n\\end{corollary}\n\n\\begin{proof}\nLet $t=\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil$, $a=m=n^{\\delta}$, and $d=2n^{1-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}$. Then,\n\\[\n\\frac{d^t}{n^{t-1}} - {n^r} \\left( \\frac{m}{n} \\right)^{t} \n= 2^{\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil} n^{\\delta} - n^r n^{-(1-\\delta)\\cdot \\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}\n\\ge 2^{\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil} n^{\\delta} - n^{\\delta}\n\\ge a.\n\\]\nNow the corollary follows from the above lemma.\n\\end{proof}\n\nThe next theorem is an easy generalization of a result of Sudakov from~\\cite{SU}.\n\\begin{theorem}\\label{thm:ub_delta}\nLet $s \\ge r\\ge 1$ and $0<\\delta<1$. Then,\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{s+r}, n^{\\delta}) < \n\\begin{cases}\nn^{2-\\frac{(1-\\delta)^2}{r-\\delta}}, & \\text{ if $\\frac{r-\\delta}{1-\\delta}$ is an integer},\\\\ \nn^{2-\\frac{(1-\\delta)^2}{r+1-2\\delta}}, & \\text{ otherwise}. \n\\end{cases}\n\\]\n\\end{theorem}\n\n\\begin{proof}\nClearly if $\\frac{r-\\delta}{1-\\delta}$ is an integer, then $n^{2-\\frac{(1-\\delta)^2}{r-\\delta}} = n^{2-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}$. Otherwise, \n\\[\nn^{2-\\frac{(1-\\delta)^2}{r+1-2\\delta}} = n^{2-(1-\\delta)\/ \\frac{r+1-2\\delta}{1-\\delta} }\n= n^{2-(1-\\delta)\/ \\left(\\frac{r-\\delta}{1-\\delta} +1\\right)}\n> n^{2-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}.\n\\]\n\nLet $G$ be a graph on $n$ vertices with at least $n^{2-(1-\\delta)\/\\lceil \\frac{r-\\delta}{1-\\delta} \\rceil}$ edges which contains no copy of $K_{s+r}$. We show that $\\alpha_s(G) \\ge n^{\\delta}$. By Corollary~\\ref{cor:sudakov} graph $G$ contains a subset of vertices $U$ of size~$n^\\delta$ such that any $W\\subseteq U$ of size $r$ has $|N(W)|\\ge n^{\\delta}$. If $G[U]$ contains no $K_s$, then $U$ is an $s$-independent set and we are done. So suppose it contains a copy of $K_s$ and denote by $W$ any $r$ vertices of such copy (recall that $r\\le s$). Clearly, $|N(W)|\\ge n^{\\delta}$. If $N(W)$ contains a copy of~$K_s$, then together with\nthe vertices in $W$ we obtain a complete subgraph of $G$ on $s+r$ vertices, a contradiction. Thus $N(W)$ is an $s$-independent set of size at least $n^{\\delta}$.\n\\end{proof}\n\n\n\n\\section{Better lower bounds on $\\textup{\\textbf{RT}}_s(n,K_{s+1}, n^{\\delta})$ for certain values of $\\delta$}\\label{sec:r_eq_1}\n\nObserve that Theorems \\ref{thm:main1}, \\ref{thm:main2} and \\ref{thm:ub_delta} (applied with $r=1$) immediately yield the following statement.\n\\begin{theorem}\\label{thm:r_eq_1}\nLet $\\varepsilon>0$ and $\\frac 12 < \\delta < 1$. Then for all sufficiently large $s$, we have \n\\[\n\\Omega\\of{n^{1+\\delta - \\varepsilon}} = \\textup{\\textbf{RT}}_s(n, K_{s+1}, n^\\delta) = O\\of{n^{1+\\delta}}.\n\\]\n\\end{theorem}\nAs it was already observed this is also optimal with respect to $\\delta$, since for $\\delta\\le 1\/2$ $\\textup{\\textbf{RT}}_s(n, K_{s+1}, n^\\delta)=0$. We will show now that for specific values of $\\delta$ one can basically remove $\\varepsilon$ from the exponent. \n\nFirst we recall some basic properties of generalized quadrangles. A \\textit{generalized quadrangle} of order $(p,q)$ is an incidence structure on a set~${P}$ of points and a set~$\\mathcal{L}$ of lines such that:\n\\begin{enumerate}[label=$({\\textup{Q}}\\textup{\\arabic*})$]\n\\item any two points lie in at most one line,\n\\item\\label{Q:2} if $u$ is a point not on a line~$ L$, then there is a unique point $w\\in L$ collinear with~$u$, and hence, no three lines form a triangle,\n\\item every line contains~$p+1$ points, and every point lies on~$q+1$ lines,\n\\item\\label{Q:4} $|P| = (pq+1)(p+1)$ and $|\\mathcal{L}| = (pq+1)(q+1)$,\n\\item\\label{Q:5} $p\\le q^2$ and $q\\le p^2$.\n\\end{enumerate}\n\n\\begin{theorem}\\label{thm:quad}\nLet $s\\ge 2$. If a generalized quadrangle of order $(p,q)$ exists, then\n\\begin{equation}\\label{thm:quad:1}\n\\textup{\\textbf{RT}}_s((pq+1)(p+1), K_{s+1}, \\Theta(pq)) = \\Theta(p^3q^2),\n\\end{equation}\nwhere the hidden constants depend only on $s$.\n\\end{theorem}\n\n\\begin{proof}\nFor a given generalized quadrangle $(P,\\mathcal{L})$ we construct the random graph $\\mathbb{G}=(V,E)$ with $V=P$ as follows. For every $L\\in \\mathcal{L}$, let $\\chi_L : L \\to {[s]}$ be a random partition of the vertices of $L$ into $s$ classes chosen uniformly at random, where the classes need not have the same size and the unlikely event that a class is empty is permitted. Next we embed the $s$-partite complete graph on $\\chi^{-1}_L (1) \\cup \\chi^{-1}_L (2) \\cup \\dots \\cup \\chi^{-1}_L (s)$. Observe that not only are $\\mathbb{G}[L]$ and $\\mathbb{G}[L']$ edge disjoint for distinct $L, L' \\in \\mathcal{L}$, but also that the partitions for $L$ and $L'$ were determined independently.\n\nObserve that by~\\ref{Q:2} $\\mathbb{G}$ is $K_{s+1}$-free and by the Chernoff bound {\\textit{w.h.p. }} $|E| = |\\mathcal{L} |\\cdot \\Omega(p^2) = \\Omega(p^3q^2)$.\n\nNow we will show that $\\alpha_s(\\mathbb{G}) \\le s^2 p q$. Consider any $C \\in {V \\choose s^2 pq}$. We will bound the probability that $\\mathbb{G}[C] \\not \\supseteq K_s$. \nFor each $L \\in \\mathcal{L}$, let $X_L$ be the event that $K_s \\not \\subseteq \\mathbb{G}[L \\cap C]$. Clearly, \n\\[\n\\Pr(X_L) \\le s \\left( 1-\\frac{1}{s}\\right)^{|L\\cap C|} \\le s e^{-|L\\cap C|\/s}\n\\]\nand by independence, \n\\[\n\\Pr \\Big( K_s \\not \\in \\mathbb{G}[C] \\Big) \n\\leq \\Pr \\Big( \\bigcap_{L \\in \\mathcal{L}} X_L \\Big) \n\\leq \\prod_{L\\in \\mathcal{L}} s e^{-|L\\cap C|\/s}\n\\leq s^{|\\mathcal{L}|} e^{-\\sum_{L\\in \\mathcal{L}} |L\\cap C|\/s }.\n\\]\nSince $\\sum_{L\\in \\mathcal{L}} |L\\cap C| = |C|(q+1)$, we get\n\\[\n\\Pr \\Big( K_s \\not \\in \\mathbb{G}[C] \\Big) \n\\leq s^{|\\mathcal{L}|} e^{-|C|(q+1)\/s}\n= e^{|\\mathcal{L}| \\log s -|C|(q+1)\/s}.\n\\]\nSo by the union bound, the probability that there exists a subset of $s^2 p q$ vertices in $V$ that contains no $K_s$ is at most\n\\[\n\\binom{|V|}{s^2 pq} e^{|\\mathcal{L}| \\log s -|C|(q+1)\/s}\n\\le |V|^{s^2 pq} e^{|\\mathcal{L}| \\log s -|C|(q+1)\/s}\n= e^{s^2 pq \\log |V| + |\\mathcal{L}| \\log s -|C|(q+1)\/s} = o(1),\n\\]\nbecause of~\\ref{Q:4} and $|C| = s^2pq$. Thus, {\\textit{w.h.p. }} $\\alpha_s(\\mathbb{G}) \\le s^2 pq$ and consequently\n\\[\n\\textup{\\textbf{RT}}_s((pq+1)(p+1), K_{s+1}, \\Theta(pq)) = \\Omega(p^3q^2).\n\\]\n\nFinally observe that the upper bound in~\\eqref{thm:quad:1} is trivial, since if $G$ is a $K_{s+1}$-free graph of order $(pq+1)(p+1)$, then $\\Delta(G) < \\alpha_s(G) = O(pq)$ yielding $|E(G)| = O(p^3q^2)$.\n\\end{proof}\n\nIt is known that when $(p,q)\\in \\{(r,r), (r,r^2),(r^2,r),(r^2,r^3),(r^3,r^2)\\} $ for any arbitrary prime power $r$, then the generalized quadrangle exists (see, e.g., ~\\cite{GO,TH}) yielding the following:\n\n\\begin{center}\n\\begin{tabular}{c | c | c | c | c }\n$p$ & $q$ & $(pq+1)(p+1)$ & $pq$ & $p^3 q^2$ \\\\\n\\hline\\hline\n$r$ & $r$ & $ \\Theta(r^3)$ & $r^2$ & $r^5$\\\\ \\hline\n$r$ & $r^2$ & $ \\Theta(r^4)$ & $r^3$ & $r^7$\\\\ \\hline\n$r^2$ & $r$ & $ \\Theta(r^5)$ & $r^3$ & $r^8$\\\\ \\hline\n$r^2$ & $r^3$ & $ \\Theta(r^7)$ & $r^5$ & $r^{12}$\\\\ \\hline\n$r^3$ & $r^2$ & $ \\Theta(r^8)$ & $r^5$ & $r^{13}$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\smallskip\n\nThus, by letting $n$ to be $\\Theta(r^3)$, $\\Theta(r^4)$, $\\Theta(r^5)$, $\\Theta(r^7)$, and $\\Theta(r^8)$, respectively, we get the following corollary.\n\\begin{corollary}\\label{cor:quad}\nLet $\\delta \\in \\{3\/5, 5\/8, 2\/3, 5\/7, 3\/4\\}$. Then\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{s+1}, \\Theta(n^{\\delta})) = \\Theta(n^{1+\\delta}).\n\\]\n\\end{corollary}\n\n\n\nIn view of Theorems~\\ref{thm:r_eq_1} and \\ref{thm:quad} and the above corollary we conjecture that actually\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{s+1}, n^{\\delta}) = \\Theta(n^{1+\\delta})\n\\] \nfor any $1\/2 < \\delta < 1$.\n\n\n\n\n\n\n\n\\section{Phase transition of $\\textup{\\textbf{RT}}_s(n, K_{2s+1}, f)$}\\label{sec:big_r}\n\nThroughout this section $\\omega = \\omega(n)$ is a function which goes to infinity arbitrarily slowly together with $n$.\n\nHere we briefly discuss an extension of a recent result of Balogh, Hu, and Simonovits~\\cite{BHS}, who showed (answering a question of Erd\\H{o}s and S\\'os~\\cite{ES}) that\n\\begin{equation}\\label{eq:BHS:1}\n\\textup{\\textbf{RT}}_2(n,K_5, \\sqrt{n\\log n} \/ \\omega) = o(n^2)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:BHS:2}\n\\textup{\\textbf{RT}}_2(n,K_5, c \\sqrt{n\\log n}) = n^2\/4 + o(n^2)\n\\end{equation}\nfor any $c>1$. Thus, one can say that $K_5$ has a (Ramsey-Tur\\'an) phase transition at $c\\sqrt{n\\log n}$ for every $c>1$. One can easily show that for any $s\\ge 2$, $K_{2s+1}$ also has a phase transition (with respect to the $s$-independence number). This will follow from the following two theorems.\n\n\\begin{theorem}\\label{thm:ub_ks_r}\nLet $s\\ge 2$. Then \n\\[\n\\textup{\\textbf{RT}}_s(n,K_{2s+1}, \\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)) = o(n^2).\n\\]\n\\end{theorem}\n\nFirst we derive another corollary from Lemma~\\ref{lem:dependent}.\n\n\\begin{corollary}\\label{cor:dependent2}\nLet $\\varepsilon>0$ be fixed and $s\\ge 1$ be an integer. Let $G$ be a graph on $n$ vertices with at least $\\varepsilon n^2$ edges. \nThen for sufficiently large $n$, $G$ contains a subset $U$ of at least $n\/\\omega$ vertices such that every $s+1$ vertices in $U$ have at least $\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$ common neighbors.\n\\end{corollary}\n\n\\begin{proof}\nLet $r=s+1$, $t=2s+1$, $a = n\/\\omega$, $m = \\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$, and $d=2\\varepsilon n$. Observe that by~\\eqref{f:1}, $\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega) = o(n^{(s+1)\/(2s+1)})$.\nThus,\n\\[\n\\frac{d^t}{n^{t-1}} - {n^r} \\left( \\frac{m}{n} \\right)^{t} \n\\ge (2\\varepsilon)^{2s+1} n - \\frac{(\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega))^{2s+1}}{n^s} \\ge n\/\\omega = a,\n\\]\nfor sufficiently large~$n$.\nNow the corollary follows from Lemma~\\ref{lem:dependent}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:ub_ks_r}]\nLet $\\varepsilon>0$ be an arbitrarily small constant and let $G$ be a $K_{2s+1}$-free graph of order $n$ with $\\varepsilon n^2$ edges and $\\alpha_s(G)< \\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$. By Corollary~\\ref{cor:dependent2} there is a set $U$ of size $n\/\\omega$ such that every $s+1$ vertices in $U$ have at least $\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$ common neighbors. First observe that $G[U]$ contains a copy of $K_{s+1}$. Otherwise, $G[U]$ is a $K_{s+1}$-free graph of order $n\/\\omega$ with $\\alpha_s(G[U])<\\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$. But clearly this cannot happen.\nNow let $W$ be the set of vertices of a copy of $K_{s+1}$ in $U$. If $N(W)$ contains a copy of $K_s$, then such a copy together with $G[W]$ gives a copy of $K_{2s+1}$. Thus, $G[N(W)]$ is $K_s$-free. But this yields that $\\alpha_s(G)\\ge |N(W)|\\ge \\textup{\\textbf{f}}_{s,s+1}(n\/\\omega)$, a contradiction.\n\\end{proof}\n\nThe following extends a result of Erd\\H{o}s, Hajnal, Simonovits, S\\'os, and Szemer\\'edi~\\cite{EHSSS} for small $s$-independence numbers.\n\n\\begin{theorem}\\label{thm:lb_ks_r}\nLet $k\\ge 2$, $s\\ge 2$, $1\\le r\\le k-1$. Let $c$ be a constant satisfying\n\\[\nc > \n\\begin{cases}\n1 \\text{ if } s=2,\\\\\nk \\text{ if } s\\ge 3.\\\\\n\\end{cases}\n\\]\nThen,\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{ks+r}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)) \\ge \\frac{1}{2} \\left(1-\\frac{1}{k} \\right) n^2 + o(n^2). \n\\]\nFurthermore, if $r=1$, then \n\\[\n\\textup{\\textbf{RT}}_s(n,K_{ks+1}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)) = \\frac{1}{2} \\left(1-\\frac{1}{k} \\right) n^2 + o(n^2).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nFirst we assume that $s\\ge 3$ and $c > k$.\nLet $H$ be a $K_{s+1}$-free graph of order $n\/k + o(n)$ with $\\alpha_s(H) < c\/k\\cdot\\textup{\\textbf{f}}_{s,s+1}(n\/k)$. The existence of such graph follows from the definition of the Erd\\H{o}s-Rogers number. Let $G=(V,E)$ be a graph of order $n$ such that $V=V_1\\cup\\dots\\cup V_{k}$, $|V_1| = \\dots = |V_k| = n\/k+o(n)$, $G[V_i]$ is isomorphic to $H$ for each $1\\le i\\le k$, and for each $v\\in V_i$ and $w\\in V_j$ we have $\\{v,w\\}\\in E$ for any $1\\le i < j\\le k$. Observe that $G$ is $K_{ks+r}$-free and $\\alpha_{s}(G) < c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)$. Thus,\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{ks+r}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)) \\ge \\binom{k}{2} \\left( \\frac{n}{k} \\right)^2 + o(n^2) = \\frac{1}{2} \\left(1-\\frac{1}{k} \\right) n^2 + o(n^2).\n\\]\n\nNow observe that if $s=2$ and $c>1$, then it suffices to assume that $\\alpha_s(H) < c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)$, since any independent set in $G$ must lie entirely in some $V_i$.\n\nFinally, if $r=1$, then\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{ks+1}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/k)) \\le \\textup{\\textbf{RT}}_s(n,K_{ks+1}, o(n)) \\le \\frac{1}{2} \\left(1-\\frac{1}{k} \\right) n^2 + o(n^2),\n\\]\nwhere the latter follows from Theorem 2.6\\,(a) in~\\cite{EHSSS}.\n\\end{proof}\n\nSince by~\\eqref{f:1b} we have $ \\textup{\\textbf{f}}_{2,3}(n\/2) \\le (1+o(1))\\sqrt{n\\log n}$ and $ \\textup{\\textbf{f}}_{2,3}(n\/\\omega) \\ge \\sqrt{n\\log n}\/\\omega$, the above theorems can be viewed as generalizations of \\eqref{eq:BHS:1} and \\eqref{eq:BHS:2}, which are the case $k=s=2$ and $r=1$.\n\nIn particular, when $k=2$ and $r=1$, we get that $\\textup{\\textbf{RT}}_s(n,K_{2s+1}, n^{\\delta}) = n^2\/4 + o(n^2)$ for any $1\/2<\\delta<1$ and $\\textup{\\textbf{RT}}_s(n,K_{2s+1}, n^{\\delta})=o(n^2)$ for $\\delta \\le 1\/2$. Also observe that if we replace $K_{2s+1}$ by $K_{2s}$, then $\\textup{\\textbf{RT}}_s(n,K_{2s}, n^{\\delta}) = o(n^2)$ for any $\\delta<1$ (by Theorem \\ref{thm:ub_delta}).\nHere we bound $\\textup{\\textbf{RT}}_s(n,K_{2s}, n^{\\delta})$ from below using the Zarankiewicz function.\nRecall that the \\emph{Zarankiewicz function} $\\textrm{\\textbf{z}}(n,s)$ denotes the maximum possible number of edges in a $K_{s,s}$-free bipartite graph $G = (U\\cup V, E)$ with $|U| = |V| = n$.\n\n\\begin{theorem}\\label{thm:2s}\nLet $s\\ge 2$ and $c>2$. Then\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{2s}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/2)) \\ge \\textrm{\\textbf{z}}(n\/2,s).\n\\]\n\\end{theorem}\n\n\\begin{proof}\nLet $H$ be a $K_{s+1}$-free graph of order $n\/2$ with $\\alpha_s(H) < c\/2 \\cdot\\textup{\\textbf{f}}_{s,s+1}(n\/2)$. Let $G=(V,E)$ be a graph of order $n$ such that $V=V_1\\cup V_2$, $|V_1| = |V_2| = n\/2$, and $G[V_1]$ and $G[V_2]$ are isomorphic to $H$. Between $V_1$ and $V_2$ we embed a $K_{s,s}$-free bipartite graph with $z(n\/2,s)$ edges. Clearly $G$ is $K_{2s}$-free and $\\alpha_{s}(G) < c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/2)$. Thus,\n\\[\n\\textup{\\textbf{RT}}_s(n,K_{2s}, c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/2)) \\ge |E| \\ge \\textrm{\\textbf{z}}(n\/2, s).\n\\]\n\\end{proof}\n\nIt is known due to a result of Bohman and Keevash~\\cite{BK} that for sufficiently large $n$ there exists a $K_{s,s}$-free graph of order~$n$ with \n$\\Omega\\left(n^{2-2\/(s+1)} \\cdot (\\log n)^{1\/(s^2-1)}\\right)$ edges. Since each graph has a bipartite subgraph with at least half of the edges, we get that \n\\[\n\\textrm{\\textbf{z}}(n, s) \\ge \\frac{1}{2}\\textrm{\\textbf{ex}}(n,K_{s,s}) = \\Omega\\left(n^{2-2\/(s+1)} \\cdot (\\log n)^{1\/(s^2-1)}\\right).\n\\]\nThis together with Theorem~\\ref{thm:2s} implies that \n\\begin{equation}\\label{eq:K2s}\n\\textup{\\textbf{RT}}_s(n,K_{2s}, f) = \\Omega\\left(n^{2-2\/(s+1)} \\cdot (\\log n)^{1\/(s^2-1)}\\right)\n\\end{equation}\nfor any $s\\ge 2$ and $f \\ge c\\;\\!\\textup{\\textbf{f}}_{s,s+1}(n\/2)$ with $c>2$.\n\n\n\\section{Concluding remarks}\n\nThe study of the Ramsey-Tur\\'an number for small $s$-independence number brings several new problems and challenges. We believe that now the most interesting question is to decide whether the lower bounds given by Theorems~\\ref{thm:main1} and~\\ref{thm:main2} are nearly optimal for any $r\\ge 2$. We believe that the upper bound in Theorem~\\ref{thm:ub_delta} can be improved.\n\nIt is also not hard to verify that the proofs of Theorems~\\ref{thm:main1} and~\\ref{thm:main2} can allow $r$ to grow with $s$. For fixed $\\varepsilon$ and $\\delta$, these theorems still hold for $r = r(s) \\le c s^{1\/5}$ for some small constant $c$ which may depend on $\\varepsilon$ and $\\delta$. Of course, since $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ is nondecreasing in $r$, we get some lower bound for larger $r$ as well, but this fact is not particularly satisfying. While we did not put much effort into allowing $r$ to grow large with $s$, new ideas would be needed to prove lower bounds on $\\textup{\\textbf{RT}}_s(n, K_{s+r}, n^\\delta)$ that get better as $\\delta$ increases and which apply to say, all $1 \\le r \\le s$. (Recall that the upper bound in Theorem~\\ref{thm:ub_delta} applies to all $1 \\le r \\le s$, and that we have the lower bound~\\eqref{eq:K2s} but this bound stays the same even for $\\delta$ quite close to $1$.)\n\n\n\n\n\n\\section{Acknowledgment}\nWe are grateful to an anonymous referee for bringing to our attention reference~\\cite{BHS}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSince the successful exfoliation of single layer graphene, 2D materials have been attracting significant interest due to their highly tunable physical properties and immense potential in scalable device applications \\cite{novoselov2004electric,novoselov2005two,zhang2005experimental,geim2007rise,RevModPhys.81.109}. However, pristine graphene exhibits no band gap and its inherent inversion symmetry suppresses the spin-orbit coupling (SOC) \\cite{zibouche2014transition, PhysRevB.92.144404, RevModPhys.76.323, macdonald2005ferromagnetic, dietl2010ten, deng2011li}. The weak SOC and zero band gap eliminate graphene as a potential candidate for being applied to spintronic devices, which require one to search for alternative 2D materials that extend beyond graphene to other layered materials with van der Waals gaps \\cite{PhysRevB.92.144404, RevModPhys.76.323, macdonald2005ferromagnetic, dietl2010ten, deng2011li}. For example, in single-layer MoS$_2$, the large SOC leads to a unique spin-valley coupling which may be useful for spintronic applications \\cite{PhysRevB.84.153402,PhysRevLett.108.196802,PhysRevB.86.165108,PhysRevB.88.125301,xu2014spin}. Whereas spintronic devices using 2D materials are still in their infancy \\cite{han2014graphene,ohno2000electric,chang2013experimental,PhysRevLett.113.137201}, which is due to the lack of long-range ferromagnetic order that is crucial for macroscopic magnetic effects \\cite{PhysRevLett.114.016603,gonzalez2016atomic}. The emergence of ferromagnetism in 2D materials in combination with their rich electrical and optical properties could open up ample opportunities for 2D magnetic, magneto-electric, and magneto-optic applications \\cite{ohno2000electric,chang2013experimental,gong2017discovery}.\n\nRecently, Chromium Tellurides Cr$X$Te$_3$ ($X$ = Si, Ge, and Sn) with the centrosymmetric have arisen significant attention because they belong to a rare category of ferromagnetic semiconductors possessing a 2D layered structure\\cite{PhysRevB.92.144404,gong2017discovery,PhysRevB.91.235425,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,PhysRevB.92.035407,siberchicot1996band,PhysRevX.3.031002,li2014crxte,PhysRevB.92.165418,PhysRevLett.117.257201,liu2016critical,carteaux19952d}. Extensive theoretical and experiment efforts have been extended towards understanding the properties of these 2D magnets. On the theoretical side, recent studies on Cr$X$Te$_3$ have been focusing on their electronic structure and magnetic properties, particularly predictions of the single-layer properties \\cite{PhysRevB.91.235425,PhysRevB.92.035407,siberchicot1996band,PhysRevX.3.031002,li2014crxte,PhysRevB.92.165418,PhysRevLett.117.257201}. On the experimental side, CrSiTe$_3$ and CrGeTe$_3$ have been prepared and characterized \\cite{PhysRevB.92.144404,gong2017discovery,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,liu2016critical,carteaux19952d}. Comparing with CrSiTe$_3$, showing characteristics of a 2D-Ising behavior\\cite{PhysRevB.92.144404,liu2016critical,carteaux19952d}, the smaller van der Waals gap and the larger in-plane nearest-neighbor Cr-Cr distance in CrGeTe$_3$ enhance the Curie temperature from 32 K for the CrSiTe$_3$ to 61 K for the CrGeTe$_3$ \\cite{PhysRevB.92.144404,carteaux1995crystallographic,PhysRevB.92.035407,li2014crxte}. In addition, theoretical investigations have suggested that the single-layer CrGeTe$_3$ presents characteristics of 2D-Ising behavior similar to CrSiTe$_3$\\cite{li2014crxte,PhysRevLett.117.257201}. By contrast, a scanning magneto-optic Kerr microscopy experiment, single-layer CrGeTe$_3$ represents a close-to-ideal 2D Heisenberg ferromagnetic system using the rigorous renormalized spin wave theory analysis and calculations\\cite{gong2017discovery}. It is known that, with the increase of the $X$ atom radius, Cr$X$Te$_3$ presents the smaller van der Waals gap and the larger cleavage energy \\cite{PhysRevB.92.144404,carteaux1995crystallographic,PhysRevB.92.035407,li2014crxte}. We suppose that Cr$X$Te$_3$ system may undergo a three dimensional (3D) magnetic phase transition from 2D with the increase of the $X$ atom radius. Therefore, a method to rapidly characterize the critical behavior of single-crystalline CrGeTe$_3$ is crucial. For this purpose, we present a detailed investigation of the critical phenomena of CrGeTe$_3$ using the initial isothermal $M(H)$ curves around the Curie temperature $T_\\mathrm{C}$. We find that the critical exponents of CrGeTe$_3$ satisfy the universality class of the tricritical mean-field theory. This indicates that the magnetic phase transition of CrGeTe$_3$ should be close to a tricritical point from 2D to 3D.\n\n\\section{Methods}\n\\label{sec:methods}\nSamples of single-crystalline CrGeTe$_3$ were prepared by the self-flux technique \\cite{ji2013ferromagnetic}. The XRD data indicated that the powders are single phase with the rhombohedral structure (see Supporting Information). We measured the heat capacity using the Quantum Design physical properties measurement system (PPMS-9T) and characterized the magnetic properties by the magnetic property measurement system (MPMS-XL5). Density functional theory (DFT) calculations were performed using Vienna {\\it Ab-initio} Simulation Package \\cite{Ref36}. We used the local density approximation \\cite{Ref37, Ref38} to treat the electron-electron exchange-correlation interactions. The electron-ion interactions are described by the potentials based on the projector augmented wave method \\cite{Ref39, Ref40}.\n\\section{Results}\nFigure~\\ref{fig:1}(a) and (b) show the temperature-dependent inverse susceptibility 1\/$\\chi(T)$ of CrGeTe$_3$ under field cooled cooling with applied magnetic field $H$ = 100 Oe, parallel to the $ab$ plane and $c$ axis, respectively. We observe a paramagnetic-ferromagnetic (PM-FM) transition that occurs at a critical temperature of 67.3 K, as determined by the derivative of the susceptibility. This temperature is consistent with the values of 61 K or 70 K reported previously \\cite{carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto}. For a FM system, the 1\/$\\chi(T)$ above can be described by the Curie-Weiss law resulting from the mean-field theory \\cite{Ref41}. The red curves showing the Curie-Weiss law are obeyed only at high temperatures. A close observation of Fig.\\ref{fig:1}(a) and (b) reveals that the curves deviate from straight lines at around 150 K, which is much higher than $T_C^\\mathrm{mag}$, indicating strong short-range FM spin interactions in CrGeTe$_3$ above $T_C^\\mathrm{mag}$. The effective magnetic moment $\\mu_\\mathrm{eff}$ is determined to be around 4.22$\\mu_\\mathrm{B}$ (parallel the ab plane) and 4.35$\\mu_\\mathrm{B}$ (parallel the c axis), which are close to the theoretical value expected for Cr$^{3+}$ of 3.87$\\mu_\\mathrm{B}$ \\cite{carteaux1995crystallographic}. The insets of Fig.\\ref{fig:1}(a) and (b) show the isothermal magnetization $M(H)$ at 5 K exhibiting a typical FM behavior with the saturation field $H_S$ of about 5 kOe (parallel to the ab plane) and 2.5 kOe (parallel to the c axis). In addition, the $M(H)$ curves show almost no coercive force for CrGeTe$_3$.\n\n\\begin{figure}[t]\n \\includegraphics[width=8cm]{Fig1.eps}\n \\caption{(a) and (b) Temperature-dependent inverse susceptibility 1\/$\\chi(T)$ of CrGeTe$_3$ under field cooled cooling with an applied magnetic field $H$ of 100 Oe, parallel to the $ab$ plane and $c$ axis, respectively. The red solid lines are the fitted results according to the Curie-Weiss law. The insets show the isothermal magnetization curves $M$($H$) at 5 K. (c) Specific heat $C_p$ as a function of $T$ for CrGeTe$_3$ and the fitted $C_V^\\mathrm{Debye}(T)$ using Eqs.\\ref{eq:1} and 2; Temperature-dependent magnetic (d) specific heat $C_\\mathrm{mag}(T)$ and (e) entropy $S_\\mathrm{mag}(T\\rightarrow\\infty)$. The blue dashed line refers to $S_\\mathrm{mag}(T)$ calculated with the magnetic moment S of Cr$^{3+}$ being 3\/2.}\n \\label{fig:1}\n\\end{figure}\n\nFigure~\\ref{fig:1}(c) shows the variation of the zero-field specific heat (SH) $C_p(T)$ with temperature. The sharp anomaly in $C_p(T)$ at 64.8 K corresponds to the Curie temperature $T_C^\\mathrm{SH}$. Since CrGeTe$_3$ is a seminconductor \\cite{carteaux1995crystallographic}, the electronic contribution to the heat capacity is not considered. The $C_\\mathrm{mag}$ can be calculated by the following equations \\cite{Ref41}:\n\\begin{equation}\n \\label{eq:1}\nC_\\mathrm{mag}(T) =C_p(T)-NC_V^\\mathrm{Debye}(T)\n\\end{equation}\nand\n\\begin{equation}\n \\label{eq:2}\nC_V^\\mathrm{Debye}(T) = 9R\\bigg(\\frac{T}{\\Theta_\\mathrm{D}}\\bigg)^3\\int_{0}^{\\Theta_\\mathrm{D}\/T} \\frac{x^4e^x}{(e^x-1)^2} dx,\n\\end{equation}\nwhere $R$ is the molar gas constant, $\\Theta_\\mathrm{D}$ is the Debye temperature, and $N$ = 5 is the number of atoms per formula unit. The sum of Debye functions accounts for the lattice contribution to the specific heat. We can extract the magnetic contribution $C_\\mathrm{mag}(T)$ from the measured specific heat of CrGeTe$_3$. The fitted $C_p(T)$ for CrGeTe$_3$ by Eqs.\\ref{eq:1} and 2 over the temperature range from about 7 to 250 K is shown by the red curve in Fig.\\ref{fig:1}(c) using the Debye temperature $\\Theta_\\mathrm{D}$ = 476.5 K. We observe a sharp peak at $T_C^\\mathrm{SH}$ of 64.8 K and there is strong dynamic short-range FM spin interactions above $T_C^\\mathrm{SH}$ (see Fig.\\ref{fig:1}(d)). The magnetic entropy $S_\\mathrm{mag}(T)$ is calculated by\n\\begin{equation}\n \\label{eq:3}\nS_\\mathrm{mag}(T) =\\int_{0}^{T}\\frac{C_\\mathrm{mag}(T)}{T}dT.\n\\end{equation}\nFig.\\ref{fig:1}(e) shows the temperature dependence of $S_\\mathrm{mag}(T)$. The entropy of CrGeTe$_3$ per mole with completely disordered spins S is\n\\begin{equation}\n \\label{eq:4}\nS_\\mathrm{mag}(T\\rightarrow\\infty)=6R\\mathrm{ln}(2\\mathrm{S}+1).\n\\end{equation}\nUsing S = 3\/2 for Cr$^{3+}$, we obtain $S_\\mathrm{mag}(T\\rightarrow\\infty)$ of 69.2 J\/(mol$\\cdot$K). However, we observe the $S_\\mathrm{mag}$ is 64.7 J\/(mol$\\cdot$K) at 150 K in Fig.\\ref{fig:1}(e), which is smaller than $S_\\mathrm{mag}(T\\rightarrow\\infty)$. Note that there is an error of about 10\\% \\cite{PhysRevB.72.174404} in our measurement due to the fitting of the optical phonon contributions at high temperatures. In spite of this small error, our result indicates the strong short-range FM spin interactions above $T_C^\\mathrm{SH}$.\n\n\\begin{figure}[t]\n \\includegraphics[width=8.5cm]{Fig2.eps}\n \\caption{(a) Initial magnetization of CrGeTe$_3$ around $T_\\mathrm{C}$; (b) Arrott plots of $M^2$ versus $H\/M$ (the $M(H)$ curves are measured at temperature intervals of 1 K and 0.5 K approaching $T_\\mathrm{C}$); (c) Normalized slopes as a function of temperature; (d) Modified Arrott plot ($M^{1\/\\beta}$ versus $(H\/M)^{1\/\\gamma}$) of isotherms with $\\beta$ = 0.24 and $\\gamma$ = 1 for CrGeTe$_3$. The red dashed line is the linear fit of isotherm at 67.9 K; (e) Temperature dependence of $M_S$ and $\\chi_0^{-1}$. The $T_\\mathrm{C}$ and critical exponents are obtained from the fitting of Eqs.S1 and S2; (f) The Kouvel-Fisher plot. The $T_\\mathrm{C}$ and critical exponents are obtained from the linear fit.}\n \\label{fig:2}\n\\end{figure}\n\nAs mentioned above, with the $X$ atom radius increases, the Cr$X$Te$_3$ compounds present the smaller van der Waals gap and the larger cleavage energy \\cite{PhysRevB.92.144404,carteaux1995crystallographic,PhysRevB.92.035407,li2014crxte}, which may induce a 3D magnetic phase transition. For the purpose of confirmation, we performed a detailed characterization of the critical phenomena using the initial isothermal $M(H)$ curves around $T_\\mathrm{C}$ for the CrGeTe$_3$, which are shown in Fig.\\ref{fig:2}(a). In the mean-field theory, the critical exponents and critical temperature can be determined from the Arrott plot with $\\beta$ of 0.5 and $\\gamma$ of 1.0 \\cite{Ref49, Ref50}. According to this method, the $M^2$ versus $H\/M$ (shown in Fig.\\ref{fig:2}(b)) should be a series of parallel straight lines in the higher field range around $T_\\mathrm{C}$ and the line at $T$ = $T_\\mathrm{C}$ should pass through the origin. Note that the lower-field data mainly represent the arrangement of magnetic domains, which should be excluded from the fitting process \\cite{Ref51}. However, all the curves in Fig.\\ref{fig:2}(b) show nonlinear behaviors having downward curvature even at high fields, which indicates an non-mean-field-like behavior. Moreover, the positive slope reveals a second-order phase transition according to the criterion proposed by Banerjee \\cite{Ref52}. As such, a modified Arrott plot should be employed to obtain the critical exponents.\n\n\nTo determine an accurate model, we obtain a modified Arrott plot following Eq.S5 for single-crystalline CrGeTe$_3$ at different temperatures. Three groups of possible exponents belonging to the 3D Heisenberg model ($\\beta$ = 0.365, $\\gamma$ = 1.386), 3D-Ising model ($\\beta$ = 0.325, $\\gamma$ = 1.24) and tricritical mean-field model ($\\beta$ = 0.25, $\\gamma$ = 1.0) exhibit nearly straight lines in the high field region \\cite{Ref45, Ref53}. We calculate their normalized slopes (NS) defined as NS =$ S(T)\/S(T_C^\\mathrm{mag}$ = 67.3 K). By comparing NS with the ideal value of unity, one can identify the most suitable model \\cite{Ref45, Ref53}. Fig.\\ref{fig:2}(c) shows the plots of NS versus $T$ employing the three different models, revealing that the tricritical mean-field model is the most appropriate to describe the critical behavior of CrGeTe$_3$.\n\nBy proper selections of $\\beta$ and $\\gamma$, one can clearly show the isotherms are a set of parallel straight lines at high fields as displayed in the Fig.\\ref{fig:2}(d). The linear extrapolation from the high-field region gives the spontaneous magnetization $M_S(T,0)$ and the initial inverse susceptibility $\\chi_0^{-1}(T,0)$ (see Fig.\\ref{fig:2}(e)) corresponding to the intercepts on the $M^{1\/\\beta}$ and $(H\/M)^{1\/\\gamma}$ axes, respectively. By fitting the data of $M_S(T,0)$ and to Eqs.S1 and S2, one obtains two new values of $\\beta$ = 0.242$\\pm$0.006 with $T_\\mathrm{C}$ = 67.95$\\pm$0.01 and $\\gamma$ = 0.985$\\pm$0.009 with $T_\\mathrm{C}$ = 67.90$\\pm$0.09. These results are again very close to the critical exponents of tricritical mean-field model. In addition, these critical exponents and $T_\\mathrm{C}$ can be obtained more accurately from the Kouvel-Fisher (KF) method \\cite{Ref54}. Hence, one can find that the temperature dependence of $M_S(dM_S\/dT)^{-1}$ and $\\chi_0^{-1}(d\\chi_0^{-1}\/dT)^{-1}$ should be straight lines with slopes 1\/$\\beta$ and 1\/$\\gamma$, respectively. As seen in Fig.\\ref{fig:2}(f), the linear fit yields the $\\beta$ of 0.240$\\pm$0.006 with $T_\\mathrm{C}$ of 67.91$\\pm$0.07 and $\\gamma$ of 1.000$\\pm$0.005 with $T_\\mathrm{C}$ of 67.88$\\pm$0.05, respectively. Remarkably, the obtained values of the critical exponents and $T_\\mathrm{C}$ using the KF method are in excellent agreement with those using the modified Arrott plot based on the tricritical mean-field model. This suggests that the estimated values are self-consistent and unambiguous.\n\nTo further validate the above critical exponents $\\beta$ and $\\gamma$, we study the relation among these exponents. According to Eq.S3, $\\delta$ can be directly estimated from the critical isotherm at $T_\\mathrm{C}$. Figure \\ref{fig:3}(a) shows the isothermal magnetization $M(H)$ at $T_\\mathrm{C}$ = 67.9 K. The inset of the same plot has been demonstrated on a log-log scale. The solid straight line with a slope 1\/$\\delta$ is the fitted result using Eq.S3. From the linear fit we obtained the third critical exponent $\\delta$ of 5.032$\\pm$0.005. Moreover, the exponent $\\delta$ can be calculated by the Widom scaling relation \\cite{Ref55,Ref56}\n\n\\begin{equation}\n \\label{eq:5}\n\\delta =1+\\gamma\/\\beta.\n\\end{equation}\nBased on the $\\beta$ and $\\gamma$ values calculated in Fig.\\ref{fig:2}(e) and (f), Eq.\\ref{eq:5} yields $\\delta$ of 5.070 $\\pm$ 0.006 and 5.167 $\\pm$ 0.006, respectively. We emphasize that these values are very close to the results from the experimental critical isothermal analysis. Therefore, the critical exponents obtained in this study basically obey the Widom scaling relation, showing that the obtained $\\beta$, $\\gamma$ and $\\delta$ are reliable.\n\nFinally, these critical exponents should follow the scaling equation (Eq.S6) in the critical region. The scaling equation indicates that $m$ versus $h$ forms two universal curves for $T > T_\\mathrm{C}$ and $T < T_\\mathrm{C}$, respectively. Based on Eq.S7, the isothermal magnetization around the critical temperatures for CrGeTe$_3$ has been plotted in Fig.\\ref{fig:3}(b). All experimental data in the higher-field region collapse onto two universal curves, in agreement with the scaling theory. The inset of Fig.\\ref{fig:3}(b) shows the corresponding log-log plot. Similarly, all the points collapse into two different curves in the higher-field region. This result shows again that the obtained results of the critical exponents and $T_\\mathrm{C}$ are valid.\n\n\\begin{figure}\n \\includegraphics[width = 8cm]{Fig3.eps}\n \\caption{(a) Isothermal $M(H)$ at $T_\\mathrm{C}$. The inset shows the alternative plot on a log-log scale and the straight line is the linear fit following Eq.S3; (b) Renormalized magnetization $m$ versus renormalized field $h$ at several typical temperatures around the $T_\\mathrm{C}$. The inset shows an alternative plot on a log-log scale; the effective exponents (c) below $T_\\mathrm{C}$ and (d) above $T_\\mathrm{C}$ as a function of the reduced temperature $\\varepsilon$.}\n \\label{fig:3}\n\\end{figure}\n\n\nTo further examine the convergence of the critical exponents, the effective exponents $\\beta_\\mathrm{eff}$ and $\\gamma_\\mathrm{eff}$ can be are obtained by Eqs.S8 and S9 for CrGeTe$_3$. As shown in Fig.\\ref{fig:3}(c) and (d), both $\\beta_\\mathrm{eff}$ and $\\gamma_\\mathrm{eff}$ show a non-monotonic variation with $\\varepsilon$ (see Eq.S4). The lowest $\\varepsilon$ ($\\varepsilon_\\mathrm{min}$) are 5.89$\\times$10$^{-3}$ and 1.47$\\times$10$^{-3}$ for $\\beta_\\mathrm{eff}$ and $\\gamma_\\mathrm{eff}$, respectively. We obtain the effective exponents $\\beta_\\mathrm{eff}$ of 0.242 and $\\gamma_\\mathrm{eff}$ of 1.069, indicating that both $\\beta_\\mathrm{eff}$ and $\\gamma_\\mathrm{eff}$ are converged when the temperature approaches $T_\\mathrm{C}$.\n\nThe experimental critical exponents of CrGeTe$_3$, as well as the theoretical values of CrSiTe$_3$, MnSi and some other manganites based on various models, are summarized in Table~\\ref{table:1}. It is seen that the critical exponents for MnSi and doped manganites are consistent with those of tricritical mean-field theory \\cite{Ref45,Ref53,Ref60,Ref61}. These compounds have the same characteristics,{\\it i.e.}, a tricritical point separating the first-order from the second-order ferromagnetic phase transitions. This phenomenon shows that the element substitution \\cite{Ref53,Ref61}, hole or electric doping \\cite{Ref60}, and external magnetic field\\cite{Ref45} can induce the tricritical behavior. However, CrGeTe$_3$ presents a second-order ferromagnetic phase transitions\\cite{PhysRevB.92.144404,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,PhysRevB.92.035407} and our results indicate that the critical behavior of CrGeTe$_3$ is close to the theoretical value of tricritical mean-field model. Comparing with CrSiTe$_3$, showing characteristic of 2D-Ising model \\cite{PhysRevB.92.144404,liu2016critical,carteaux19952d}, the smaller van der Waals gap and the larger planar nearest-neighbor Cr-Cr distance of CrGeTe$_3$ enhances the Curie temperature from 32 K for the CrSiTe$_3$ to 61 K for the CrGeTe$_3$ \\cite{PhysRevB.92.144404,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,PhysRevB.92.035407}. In addition, the neutron scattering and isothermal magnetization experiments yield a critical exponent $\\beta$ of around 0.151 or 0.17 for CrSiTe$_3$ \\cite{PhysRevB.92.144404,liu2016critical,carteaux19952d}, which is close to the value expected for a 2D transition ($\\beta_\\mathrm{2D}^\\mathrm{Ising}$= 0.125) and well below the values expected for a 3D transition ($\\beta_\\mathrm{2D}^\\mathrm{Ising}$ = 0.326). Our results yield a critical exponent $\\beta$ of 0.24 for CrGeTe$_3$ that is close to the critical exponent of the tricritical mean-field model. Hence, the increase of the $X$ atom radius, facilitating super exchange coupling between the Cr atoms via the Te atom and leading to the smaller van der Waals gap in Cr$X$Te$_3$ system \\cite{PhysRevB.92.144404,carteaux1995crystallographic,ji2013ferromagnetic,tian2016magneto,PhysRevB.92.035407}, could induce a tricritical magnetic phase transition in the CrGeTe$_3$ single crystal.\n\n\n\\begin{table*}[tb]\n \\caption{Critical exponents of CrGeTe$_3$ with various theoretical models, CrSiTe$_3$ and other related materials with tricritical mean-field model (SC = single crystal; PC = polycrystalline; cal = calculated from Eq.\\ref{eq:5}).}\n \\begin{ruledtabular}\n \\begin{center}\n \\begin{tabular}{ccccccc}\n Composition & Referecne & $T_\\mathrm{C}(\\mathrm{K})$ & Technique & $\\beta$ & $\\gamma$ & $\\delta$\\\\\n \\hline\n CrGeTe$_3^\\mathrm{SC}$ & This work & 67.9 & Modified Arrott plot &0.242$\\pm$0.006 &0.985$\\pm$0.003&5.070$\\pm$0.006$^\\mathrm{cal}$\\\\\n & & & Kouvel-Fisher method &0.240$\\pm$0.006 &1.000$\\pm$0.005&5.167$\\pm$0.006\\\\\n & & & Critical isotherm & & &5.032$\\pm$0.005$^\\mathrm{cal}$\\\\\n Tricritical mean-field & [\\onlinecite{Ref52}] & & Theory &0.25 & 1 &5\\\\\nMean-field & [\\onlinecite{Ref49}][\\onlinecite{Ref50}] & & Theory &0.5 & 1 &3\\\\\n3D-Heisenberg theory & [\\onlinecite{Ref49}][\\onlinecite{Ref50}] & & Theory &0.365 & 1.386 &4.8\\\\\n3D-Ising & [\\onlinecite{Ref49}][\\onlinecite{Ref50}] & & Theory &0.325 & 1.24 &4.82\\\\\n CrSiTe$_3^\\mathrm{SC}$ & [\\onlinecite{liu2016critical}] & 31 & Modified Arrott plot &0.170$\\pm$0.008 &1.532$\\pm$0.001&10.012$\\pm$0.047$^\\mathrm{cal}$\\\\\n MnSi$^\\mathrm{SC}$ & [\\onlinecite{Ref45}] & 30.5 & Modified Arrott plot &0.242$\\pm$0.006 &0.915$\\pm$0.003&4.734$\\pm$0.006\\\\\nLa$_{0.1}$Nd$_{0.6}$Sr$_{0.3}$MnO$_3$$^\\mathrm{PC}$ & [\\onlinecite{Ref53}] & 249.3 & Modified Arrott plot &0.257$\\pm$0.005 &1.12$\\pm$0.03&5.17$\\pm$0.02\\\\\nLa$_{0.9}$Te$_{0.1}$MnO$_3$$^\\mathrm{PC}$ & [\\onlinecite{Ref60}] & 239.5 & Modified Arrott plot &0.201$\\pm$0.003 &1.27$\\pm$0.04&7.14$\\pm$0.04\\\\\nLa$_{0.6}$Ca$_{0.4}$MnO$_3$$^\\mathrm{PC}$ & [\\onlinecite{Ref61}] & 265.5 & Modified Arrott plot &0.25$\\pm$0.03 &1.03$\\pm$0.05&5.0$\\pm$0.8\\\\\n\n \\end{tabular}\n \\end{center}\n \\end{ruledtabular}\n \\label{table:1}\n\\end{table*}\n\n\nAlthough single-crystalline CrSnTe$_3$ has not yet been synthesized, we speculate that the magnetism of CrSnTe$_3$ should be closer to the 3D-Ising model. To support this assumption, we perform DFT calculations with the same calculation parameters that were used in Ref.[\\onlinecite{Ref62}]. Figure~\\ref{fig:4}(a) shows the calculated formation energy $E_\\mathrm{f}$, which is defined as the energy cost of extracting a sheet of single-layer Cr$X$Te$_3$ from their bulk counterparts. As can be seen, $E_\\mathrm{f}$ increases as the species vary from Si to Ge. This is consistent with the larger theoretical cleavage energy of single-layer CrGeTe$_3$ than that of CrSiTe$_3$, which indicates that the layers are coupled more strongly in CrGeTe$_3$\\cite{li2014crxte}. The formation energy of CrSnTe$_3$ is even higher than the other two compounds, revealing that it presents the strongest interlayer coupling, which leads to its 3D characteristics. Figure~\\ref{fig:4}(b), (c), and (d) illustrate the charge density of the three compounds. Consistent with the trend of the $E_\\mathrm{f}$ results, the electron density around the Sn-Sn pair is the least among the three materials. Namely, more electrons in CrSnTe$_3$ participate the interlayer coupling.\n\\begin{figure}[t]\n\\center\n \\includegraphics[width=8.5cm]{Fig4.eps}\n \\caption{(a) Formation energy of single-layer Cr$X$Te$_3$; The formation energy of single-layer CrSiTe$_3$ is adopted from Ref.[\\onlinecite{Ref62}] (b), (c), and (d) charge density of bulk Cr$X$Te$_3$ with an isosurface value of 0.05 $e$\/$r_\\mathrm{Bohr}^3$.}\n \\label{fig:4}\n\\end{figure}\n\n\\section{Conclusions}\nIn conclusion, we have performed a comprehensive experimental study on the critical properties of single-crystalline CrGeTe$_3$ using isothermal magnetization around the Curie temperature $T_\\mathrm{C}$. Based on various experimental techniques including the modified Arrott plot, Kouvel-Fisher method and critical isotherm analysis, we obtained the critical exponents $\\beta$, $\\gamma$, and $\\delta$ of 0.240 $\\pm$ 0.006, 1.000 $\\pm$ 0.005, and 5.070 $\\pm$ 0.006, respectively, at the Curie temperature of 67.9K. These numerical results are similar to the theoretical values in the tricritical mean-field model, which is therefore capable of describing the critical magnetic behavior of 2D CrGeTe$_3$. DFT calculations show that the formation energy of CrGeTe$_3$ lies between those of CrSiTe$_3$ and CrSnTe$_3$, which is in line with a crossover of the magnetic phase transition from 2D to 3D. Overall, our findings provide a fundamental understanding of the anomalous PM-FM transition in a novel 2D ferromagnetic semiconductor.\n\n\\begin{acknowledgments}\nThis work was supported by the National Key Research and Development Program under contracts 2016YFA0300404 and 2016YFA0401003, the Joint Funds of the National Natural Science Foundation of China and the Chinese Academy of Sciences' Large-Scale Scientific Facility under contract U1432139 and U1532153, the National Natural Science Foundation of China under contract 11674326, Key Research Program of Frontier Sciences, CAS (QYZDB-SSW-SLH015), and the Nature Science Foundation of Anhui Province under contract 1508085ME103. J. Z. is supported by the Nature Science Foundation of China (Grant No. 51602079) and the Fundamental Research Funds for the Central Universities of China (Grant No. 372 AUGA5710013115). This research also used computational resources of the National Supercomputing Center of China in Shenzhen (Shenzhen Cloud Computing Center).\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\label{sec:intro}Introduction}\n\nThere is at least one Higgs boson~\\cite{LHC}; maybe there are more.\nMulti-Higgs doublet models contain new sources of CP violation, which is one\nof the required ingredients~\\cite{Sakharov} for baryogenesis. It is therefore\ninteresting to consider whether CP violation from the Higgs sector could be\nused to generate the baryon asymmetry of the Universe~\\cite{BAU,BAU2}. This\ncan occur in electroweak baryogenesis scenarios \\cite{BAU@EPT}; here we are\ninterested in asymmetries produced before the electroweak phase transition\n(EWPT).\n\nIn this paper we consider two-Higgs doublet models (2HDM)~\\cite{hhg,2HDM}. If\ninteractions which exchange Higgs flavour are sufficiently weak, then the two\npopulations of Higgs fields could contain independent asymmetries in the\nearly Universe. Since at least one of the Higgs must couple to Standard Model\n(SM) fermions, its asymmetry is redistributed among other SM particles by\nYukawa interactions, prior to the electroweak phase transition.\nHowever, hypercharge neutrality of the Universe relates the asymmetries among\nall charged particles. This implies that a relative asymmetry among the Higgs\nscalars, generated by out-of-equilibrium CP-violating processes in the Higgs\nsector, could be transformed into a baryon asymmetry in the presence of\n($B+L$)-violating sphalerons~\\cite{sphalerons}. The interest of such\nbaryogenesis scenarios is that they require no $B$ or $L$-violating\ninteractions beyond the non-perturbative sphalerons of the SM, relying only on\nCP violation in an extended Higgs sector.\n\nThe issue of ``basis-independence'' is of particular\nimportance~\\cite{Lavoura:1994fv,DH}. The point is that physical observables\ncannot depend on a basis choice in the Lagrangian ---one may ask, for\ninstance, what $\\phi_1$ and $\\phi_2$ are in the 2HDM. Clearly, the survival of\na relative asymmetry between the $\\phi_1$'s and $\\phi_2$'s in early Universe\nwill depend on the speed of interactions that exchange $\\phi_1$ with $\\phi_2$.\nBut the pertinent coupling constants naively appear to depend on the choice of\n$\\phi_1$ and $\\phi_2$. We show that such washout interactions are controlled\nby the misalignment among different couplings, and can be parameterised in a\nbasis-independent way.\n\nThe paper is organized as follows. A compendium of relevant results for the\n2HDM is given in section~\\ref{sec:2HDM}, followed by some estimates for\ninteraction rates in the early Universe. Section~\\ref{sec:TE} constrains\nparameters of the Higgs potential by requiring Higgs flavour exchange to be\nout of equilibrium. In the second part of this section, we discuss the\nbasis-independence of these bounds. In Section~\\ref{sec:chemeq} we derive the\nequations of chemical equilibrium~\\cite{KS}, which relate the asymmetries\namong SM particles and Higgs fields, due to the interactions which are in\nequilibrium. As a result, a nonvanishing equilibrium baryon asymmetry is\nobtained in the presence of a relative Higgs asymmetry, even with $B-L$\nconservation. Simple scenarios based on the out-of-equilibrium decay of\nsinglet scalar fields into Higgs doublets are presented in\nSection~\\ref{sec:scenarios}. Finally, our conclusions are summarized in\nSection~\\ref{sec:summary}.\n\n\\section{The 2HDM at finite temperature}\n\\label{sec:notn}\n\n\\subsection{Notation and review}\n\\label{sec:2HDM}\n\nThe interaction Lagrangian for the general 2HDM~\\cite{hhg,2HDM} consists of\na scalar potential plus Yukawa coupling terms. The most general gauge\ninvariant scalar potential can be written as\n\\begin{eqnarray}\n\\label{pot}\nV&=& m_{11}^2\\phi_1^\\dagger\\phi_1+m_{22}^2\\phi_2^\\dagger\\phi_2\n-[m_{12}^2\\phi_1^\\dagger\\phi_2+{\\rm h.c.}]\\nonumber\\\\[6pt]\n&&\\quad +\\frac{1}{2}\\lambda_1(\\phi_1^\\dagger\\phi_1)^2\n+\\frac{1}{2}\\lambda_2(\\phi_2^\\dagger\\phi_2)^2\n+\\lambda_3(\\phi_1^\\dagger\\phi_1)(\\phi_2^\\dagger\\phi_2)\n+\\lambda_4(\\phi_1^\\dagger\\phi_2)(\\phi_2^\\dagger\\phi_1)\n\\nonumber\\\\[6pt]\n&&\\quad +\\left[\\frac{1}{2}\\lambda_5(\\phi_1^\\dagger\\phi_2)^2\n+\\lambda_6(\\phi_1^\\dagger\\phi_1)(\\phi_1^\\dagger\\phi_2)\n+\\lambda_7(\\phi_2^\\dagger\\phi_2)(\\phi_1^\\dagger\\phi_2)+{\\rm H.c.}\\right]\\,,\n\\end{eqnarray}\nwhere $\\phi_1$ and $\\phi_2$ are two complex SU(2)$_{L}$ doublet scalar fields\nof unit hypercharge; $m_{11}^2$, $m_{22}^2$, and $\\lambda_1\\ldots\\lambda_4$\nare real parameters, while $m_{12}^2$ and $\\lambda_5,\\lambda_6,\\lambda_7$ can\nbe complex. In general, both $\\phi_1$ and $\\phi_2$ can have Yukawa couplings\nto all the SM fermions. The Yukawa interactions are\n\\begin{eqnarray}\n- {\\cal L}_Y &=&\n\\overline{Q_L}\\, (\\mathbf{\\Gamma}_1 \\phi_1 + \\mathbf{\\Gamma}_2 \\phi_2)\\, d_R\n+ \\overline{Q_L}\\, (\\mathbf{\\Delta}_1 \\tilde{\\phi}_1 + \\mathbf{\\Delta}_2\n\\tilde{\\phi}_2)\\, u_R +\n\\nonumber\\\\\n&& +\\, \\overline{L_L}\\, (\\mathbf{\\Pi}_1 \\tilde{\\phi}_1 + \\mathbf{\\Pi}_2\n\\tilde{\\phi}_2)\\, \\ell_R + \\textrm{H.c.}, \\label{Yuk_Z2}\n\\end{eqnarray}\nwhere $Q_L = (u_L, d_L)^T$ ($u_R$ and $d_R$) is a vector in the 3-dimensional\ngeneration space of left-handed quark doublets (right-handed charge $+2\/3$ and\n$-1\/3$ quarks). Accordingly, $\\mathbf{\\Gamma}_1$, $\\mathbf{\\Gamma}_2$,\n$\\mathbf{\\Delta}_1$, and $\\mathbf{\\Delta}_2$ are $3 \\times 3$ matrices in the\nrespective quark generation spaces. Similarly, $L_L = (\\nu_L, \\ell_L)^T$ and\n$\\ell_R$ are vectors in the 3-dimensional generation space of left-handed\nlepton doublets and right-handed charged leptons, respectively, while\n$\\mathbf{\\Pi}_1$ and $\\mathbf{\\Pi}_2$ are $3 \\times 3$ matrices. For\nsimplicity, we assume that there are no right-handed neutrino fields.\n\nUnder global SU(2) transformations in $(\\phi_1, \\phi_2)$ space, the kinetic\nterms of the Higgs doublets are invariant, whereas the parameters of the\nscalar potential (and the Yukawa couplings) will be modified. Such basis\ntransformations in the Lagrangian cannot affect observables, so the numerical\nvalue of the parameters in Eq.~(\\ref{pot}) is only meaningful when the basis\nis specified. Three obvious Higgs basis choices can be envisaged:\n\\begin{itemize}\n\\item $m_{12}^2 = 0$ basis, where we put a tilde on the parameters\n ($\\tilde{\\lambda}_i, \\tilde{m}_{ii}^2$, $\\tilde{y}^f_i$),\n\\item symmetry basis, where the parameters are lower case with a prime\n ($\\lambda'_i, m_{ij}^{'2}$, ${y'}^f_i$),\n\\item (thermal) mass eigenstate basis, where the parameters are uppercase\n ($\\Lambda_i, M_{ij}$, ${Y}^f_i$).\n\\end{itemize}\nHere, $\\tilde{y}^f_i$, ${y'}^f_i$ and ${Y}^f_i$ denote the Yukawa matrices of\nthe SM fermions $f$ interacting with the Higgs $i$, in the corresponding Higgs\nbasis (so $y^u_i = (\\Delta_1,\\Delta_2)$, and so on).\n\nSince our goal is to store an asymmetry between the Higgs populations prior to\nthe EWPT, interactions which exchange $\\phi_1 \\leftrightarrow \\phi_2$ must be\nsmall (see next section). We refer to such interactions as (Higgs)\nflavour-exchanging processes. For instance, in the $m_{12}^2 = 0$ basis, the\noffending parameters from the potential are $\\tilde{\\lambda}_5$,\n$\\tilde{\\lambda}_6$ and $\\tilde{\\lambda}_7$. In the Yukawa sector,\ninteractions of both Higgs doublets to either quarks or leptons will be\nstrongly constrained. This is because a relative asymmetry in the two Higgs\npopulations should be preserved, so the two Higgs fields cannot both share\ntheir asymmetry with the same fermions.\n\nSome of the undesirable couplings can be suppressed by imposing a discrete\n$Z_2$ symmetry\n\\begin{equation} \\phi_1 \\to -\\phi_1\\,,\\quad \\phi_2 \\to \\phi_2\\,.\\label{Z2}\n\\end{equation}\nIn the basis where the symmetry has the above form, it implies $m'_{12}\n=\\lambda'_6 = \\lambda'_7 = 0$, so the Higgs sector contains no explicit CP\nviolation, because the phase of $\\lambda'_5$ can be rotated away by a phase\nchoice of the Higgs fields. If both scalar fields couple to fermions of the\nsame charge, then there will be flavour changing neutral scalar interactions,\nwhich are strongly constrained by experiment. The undesirable Yukawa\ninteractions can be removed by extending the $Z_2$ symmetry of Eq.~(\\ref{Z2})\nto the fermion sector, so that each fermion charge sector only couples to one\nof the Higgs scalars. The four ways to implement this symmetry are shown in\nTable~\\ref{table1}.\n\\begin{table*}[t!]\n\\centering\n\n\\begin{tabular}{|c|cc|}\n\\hline\nModel type & $\\phi_1$ & $\\phi_2$\\\\\n\\hline\nType I & & $u$, $d$, $\\ell$\\\\\nType II & $d$, $\\ell$ & $u$ \\\\\nType X & $\\ell$ & $u$, $d$ \\\\\nType Y & $d$ & $u$, $\\ell$\\\\\n\\hline\n\\end{tabular}\n\\caption{The four types of $Z_2$ models and the corresponding Higgs couplings\nto fermions. Type X is also known as ``lepton specific'', and type Y as\n``flipped''. In the usual $\\phi_{u,d}$ notation, $\\phi_u = \\phi_2$\nalways.}\n\\label{table1}\n\\end{table*}\n\n\nThe discovery of a 125 GeV scalar at LHC places constraints on the 2HDM\nparameter space, studied so far in the context of a $Z_2$ symmetry\n($\\lambda'_6 = \\lambda'_7 = 0$), occasionally exact ($m'_{12} =\n0$)~\\cite{reviews}. After electroweak symmetry breaking, the neutral\ncomponents of the scalar fields acquire the vacuum expectation values (VEVs)\n$\\langle \\phi_1^0 \\rangle = v_1$ and $\\langle \\phi_2^0 \\rangle = v_2$, where\n$v = (v_1^2 + v_2^2)^{1\/2} \\simeq 174$~GeV. Of the eight components in the\ntwo Higgs doublets, three provide longitudinal components to $W^\\pm$ and $Z$,\ntwo create a $H^\\pm$ pair, two yield the Higgs scalar ($m_h = 125$ GeV) and\nanother neutral scalar ($H$), and the last gives a pseudoscalar ($A$). These\nmasses and the VEVs are related to the parameters of the potential. In\nparticular, \\textit{if both $v_1 \\neq 0$ and $v_2 \\neq 0$}, then the\nstationarity conditions can be used to write the pseudoscalar mass\n\\begin{equation}\nm_A^2 = \\frac{v^2}{v_1 v_2}\\, m^{'2}_{12} - 2\\lambda'_5\\, v^2.\n\\end{equation}\nThis shows that requiring small $\\phi_1 \\leftrightarrow \\phi_2$ exchanges\nthrough $m'_{12} \\sim 0$ and $\\lambda'_5 \\sim 0$, leads to $m_A \\sim 0$,\nunless $v_1 v_2 \\sim 0$. This occurs because, in the $m'_{12} = \\lambda'_5 =\n\\lambda'_6 = \\lambda'_7 = 0$ limit, the potential in Eq.~\\eqref{pot} has a\nglobal $U(1)$ symmetry, which is broken by $v_1 v_2 \\neq 0$, with the\nconsequent appearance of a massless Goldstone boson ($m_A=0$).\n\nOne solution is to consider the inert model~\\cite{inert}, which is a Type I\n2HDM with exact $Z_2$ and $v_1 = 0$\\footnote{In the usual notation for the\ninert doublet model, only $\\phi_1$ couples to fermions, while $v_2=0$. In the\nnotation used here, the role of $\\phi_1$ and $\\phi_2$ are reversed, implying\nthe changes $m'_{11} \\leftrightarrow m'_{22}$ and $\\lambda'_1 \\leftrightarrow\n\\lambda'_2$.}. In that case,\n\\begin{equation}\nm_A^2 = m^{'2}_{22} + (\\lambda'_3 + \\lambda'_4- \\lambda'_5)\\, v^2.\n\\label{mA_inert}\n\\end{equation}\nThis mass can be kept nonzero, even if $\\lambda'_5 = 0$, because the vacuum\nwith $v_1=0$ does not break the global $U(1)$. The only consequence of\n$\\lambda'_5 = 0$ is $m_H=m_A$. Because the inert $\\phi_1$ does not couple to\nfermions, the lightest particle is a candidate for dark matter. A series of\nvery clear analyses of this model, including constraints from both LHC and\nWMAP, have been performed by the Warsaw group~\\cite{warsaw}. They find large\nregions of parameter space consistent with all known data, especially if the\n$h \\rightarrow \\gamma \\gamma$ signal is consistent with the SM\n($R_{\\gamma\\gamma} \\sim 1$). This is within the 2$\\sigma$ ranges of current\nATLAS~\\cite{Aad:2013wqa} and CMS~\\cite{CMS_gg} data\n\\begin{eqnarray}\n\\textrm{ATLAS:}\n&\\ \\ \\ &\nR_{\\gamma\\gamma} = 1.55^{+0.33}_{-0.28}\\,,\n\\nonumber\\\\\n\\textrm{CMS:}\n&\\ \\ \\ &\nR_{\\gamma\\gamma} = {0.78}^{+0.28}_{-0.26}\\,.\n\\end{eqnarray}\nValues of $R_{\\gamma\\gamma}$ larger than one restrict considerably the\nparameter space.\n\nAn additional constraint imposed on the Higgs spectrum by our baryogenesis\nscenario is that $\\phi_1$ and $\\phi_2$ should be present in the thermal bath\nuntil the EWPT. If one of the $\\phi_i$ is sufficiently heavy that its\npopulation decays away prior to the EWPT, then the relative Higgs asymmetry\nis lost.\n\n\\subsection{Thermal masses and interaction rates}\n\nIn this section we review interaction rates in a thermal bath. The relevant\neigenbasis for the external leg particles should be the thermal mass\neigenstate basis, so we start by estimating the thermal mass matrix of the\nHiggs scalars, at temperatures $T \\gg |m_{ij}|$. At finite temperature, the\nlowest order contribution to the mass-squared matrix is~\\cite{CE}\n\\begin{equation}\nm^2_{ij}(T) \\simeq\n\\frac{\\partial^2 V_{eff}(T, \\phi_k)}{\\partial\\phi_i \\partial\\phi^*_j},\n\\label{defnmT}\n\\end{equation}\nwhere $V_{eff}(T, \\phi_k)= V + V_T$ is the effective potential,\nwith $V$ given in Eq.~(\\ref{pot}).\nIn the high temperature limit ($T \\gg |m_{ij}|$),\n\\begin{equation}\nV_T = \\frac{T^2}{12} {\\rm Tr} \\big[ \\mathbf{m}^2 \\big] =\n\\frac{T^2}{24}\\sum_{i=1,2} g_i M^2(\\phi_i)\\,,\n\\label{VT}\n\\end{equation}\nwhere $g_i = 4$ for a complex doublet field, and the trace is calculated\nover the $T=0$ scalar mass-squared matrix in background field, {\\it i.e.}\nallowing $\\phi_1$ and $\\phi_2$ to have non-zero values so that $M(\\phi_i)$\nare field-dependent masses.\n\nNeglecting zero-temperature loop contributions and finite-temperature fermion\nand gauge contributions, we find\n\\begin{equation}\nV_T =\\frac{T^2}{12}\\left\\{\n 3\\lambda_1|\\phi_1|^2 +(2 \\lambda_{3} + \\lambda_{4})(|\\phi_2|^2+|\\phi_1|^2)\n+ 3\\lambda_2|\\phi_2|^2\n-\n[3(\\lambda_6 +\n \\lambda_7)\\phi^{\\dagger}_1 \\phi_2\n+{\\rm H.c.}]\\right\\},\n\\end{equation}\nwhere the $m_{ij}^2$ terms are dropped because they give no contribution to\nEq.~(\\ref{defnmT}). For an arbitrary basis in the Higgs doublet space~\\cite{Ivanov},\nthis gives the thermal mass-squared matrix\n\\begin{eqnarray}\n\\mathbf{m}^2(T)& \\simeq&\n\\frac{T^2}{12}\\left(\n\\begin{array}{cc}\n3\\lambda_1 + 2 \\lambda_{3} + \\lambda_{4}&\n-3\\lambda_{6} - 3\\lambda_{7}\\\\\n -3\\lambda^*_{6} - 3\\lambda^*_{7}\n & 3\\lambda_2 + 2 \\lambda_{3} + \\lambda_{4}\\\\\n\\end{array}\n\\right)\n+\n\\left(\n\\begin{array}{cc}\n m_{11}^2&m_{12}^2 \\\\\nm_{12}^{*2} &m_{22}^2\n\\end{array}\n\\right).\n\\label{mtherm}\n\\end{eqnarray}\nDiagonalising this matrix gives the thermal mass eigenstate basis. In the\npresence of a $Z_2$ symmetry, the thermal masses are simply given by\n\\begin{eqnarray}\nm_{11}^2(T) &=& \\frac{T^2}{12} (3 \\lambda_1 + 2 \\lambda_3 + \\lambda_4) + m_{11}^2,\n\\nonumber\\\\\nm_{22}^2(T) &=& \\frac{T^2}{12} (3 \\lambda_2 + 2 \\lambda_3 + \\lambda_4) + m_{22}^2,\n\\end{eqnarray}\nso that no term of the type $m_{12}^2(T)\\, \\phi_1^\\dagger \\phi_2$ is\ngenerated. Therefore, in the latter case, the only link between $\\phi_1$ and\n$\\phi_2$ in the Higgs potential comes from the $\\lambda_5$ term, both at zero\nand at finite temperature.\n\n\nWe now review the assumptions and approximations involved in our estimates for\nthe interaction rates. We take ``thermal equilibrium\" to describe a particle\nspecies distributed following a Maxwell-Boltzmann distribution. At\ntemperatures $T \\ll m_{\\rm GUT} \\simeq 10^{16}$~GeV, this will be the case for\nparticles with SM gauge interactions. We define an interaction to be in\n``chemical equilibrium'' if it is fast enough to impose relations among the\nasymmetries in the participating particles. This will be the case if its\ntimescale, $1\/\\Gamma$, is much shorter than the age of the Universe $\\sim\n1\/H$, i.e. $\\Gamma \\gg H$, where\n\\begin{equation}\nH = \\sqrt{\\frac{4\\pi^3 g_\\ast}{45}} \\frac{T^2}{m_P} \\simeq \\frac{\n17\\, T^2}{m_P}, \\label{hubble}\n\\end{equation}\nis the Hubble expansion parameter, $g_\\ast$ is the number of\nrelativistic degrees of freedom ($g_\\ast=107.75$ in the 2HDM) and $m_P = 1.22\n\\times 10^{19}$~GeV is the Planck mass.\n\nWe estimate the interaction rate $\\Gamma = \\gamma\/n$, where $n_i \\simeq\ng_iT^3\/\\pi^2$ is the equilibrium density of an incident (massless) particle,\nand $\\gamma$ is the interaction density. For a process $ij \\to mn$, where\nall the participating particles are in thermal equilibrium, $\\gamma$ is the\nthermally averaged scattering rate,\n\\begin{eqnarray} \\label{gammarate}\n\\gamma(ij &\\to& mn)\n= \\langle n_i n_j \\sigma(i+j \\to..) \\rangle \\nonumber\\\\\n && = \\int d\\Pi_i d\\Pi_j f^{eq}_i f^{eq}_j\n\\int |{\\cal M}(i+j \\rightarrow m+n )|^2\n~(2\\pi)^4{\\delta}^4( p_i+p_j - p_m - p_n)~\nd\\Pi_m d\\Pi_n \\nonumber \\\\\n&&= \\frac{g_1 g_2 |\\Lambda|^2\\, T^4}{ 32\\pi^5},\n\\label{gamlam}\n\\end{eqnarray}\nwhere $g_i$ is the number of degrees of freedom of the particle in the bath\n(2 for a doublet), $d\\Pi = \\dfrac{d^3p}{2E(2 \\pi)^3}$ is the relativistic\nphase space, and $f^{eq}$ is the Maxwell-Boltzmann equilibrium distribution.\nThe last equality in Eq.~\\eqref{gammarate} is the result for $|{\\cal M}(i+j\n\\rightarrow m+n )|^2 = |\\Lambda|^2$.\n\n\\section{Keeping Higgs flavour-exchanging interactions out of equilibrium}\n\\label{sec:TE}\n\nWe suppose that particle-antiparticle asymmetries in $\\phi_1$ and $\\phi_2$\nwere generated at some earlier epoch of the Universe. In\nsection~\\ref{sec:scenarios}, we shall illustrate this in a simple framework.\nWe focus on the relative asymmetry between the two Higgs doublets:\n\\begin{equation}\n\\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\equiv\n\\frac{n_{\\phi_1}- n_{\\overline{\\phi}_1}}{s}- \\frac{n_{\\phi_2}-\nn_{\\overline{\\phi}_2}}{s}, \\label{asym}\n\\end{equation}\nwhere $s$ is the entropy density of the Universe. We use the notation\n$\\mathcal{Y}_{\\Delta X}$ for the asymmetry $\\mathcal{Y}_X -\n\\mathcal{Y}_{\\overline{X}}$, where $\\mathcal{Y}_X = n_X\/s$ is the comoving number\ndensity.\nThis asymmetry will be conserved as long as Higgs flavour-exchanging\ninteractions are out of equilibrium. In this section, we identify these\ninteractions, estimate the constraints on the couplings, and express these\nbounds in some useful bases.\n\nIn the thermal mass eigenstate basis, the flavour-exchanging Higgs\ninteractions that must be out of equilibrium are mediated by the quartic\ncouplings $\\Lambda_5$, $\\Lambda_6$, and $\\Lambda_7$. Requiring $ \\Gamma \\ll H$\nat $T \\simeq 100$~GeV, and using Eqs.~(\\ref{hubble}) and (\\ref{gamlam}),\nimplies\n\\begin{equation}\n|\\Lambda_n|\n\\lesssim {\\rm few} \\times 10^{-7},\\quad n =5,6,7,\n\\label{bdL}\n\\end{equation}\nto keep the Higgs asymmetries separate for temperatures down to the EWPT. This\ncondition applies in the thermal mass eigenstate basis; we translate it below\nto other bases.\n\nHiggs flavour could also be exchanged via Yukawa couplings, if both Higgs\ndoublets interact with the same fermions. For simplicity, we only consider\nthe third generation of fermions. The $t$, $b$ and $\\tau$ have Yukawa\ninteractions to both Higgs fields, so their Yukawa couplings are vectors in\nHiggs doublet space, which we represent capitalised in the thermal mass\neigenstate basis. For instance, the top Yukawa coupling is $(Y^t_1, Y^t_2)$,\nwith $m_t = Y^t_1\\, v_{1}^{T} + Y^t_2\\,v_{2}^{T}$, where the $v_{i}^{T}$ are\nthe zero-temperature Higgs VEVs in the thermal basis. The survival of the\nrelative Higgs asymmetry requires that the Yukawa interactions, between a\nfermion species $f = t,b,\\tau$ and one of the Higgs doublets, be out of\nequilibrium:\n\\begin{equation}\n\\min_i \\gamma(f+g\/\\gamma \\to f+\\phi_i) \\ll n_{f} H.\n\\end{equation}\nFor instance, in the case of the top quark, this gives\n\\begin{equation}\n \\theta_t^2 \\frac{\\alpha_s}{16\\pi^2} \\frac{m_t^2}{v^2} \\ll 17 \\frac{T}{m_P},\n\\label{topbd}\n\\end{equation}\nwhere $\\theta_t$ is the rotation angle between the thermal mass eigenstate\nHiggs basis and the eigenvector of the top Yukawa coupling (cf.\nAppendix~\\ref{sec:appendix}):\n\\begin{equation} \\label{thetatapprox}\n\\theta_t \\simeq \\frac{|Y^t_1\\,Y^t_2|}{\\sqrt{|Y^t_1|^2 + |Y^t_2|^2}}\\,\n\\simeq\n\\frac{\\left|m_{12}^2(T)\\right|}{\\left|m_{11}^2(T) - m_{22}^2(T)\\right|}\n\\,,\n\\end{equation}\nwhere the last expression is in the Yukawa eigenbasis.\n\nAt $T \\simeq 100$~GeV, Eq.~\\eqref{topbd} requires\n\\begin{equation}\\label{thetatbound}\n\\theta_t\\lesssim {\\rm few}\\times 10^{-7}.\n\\end{equation}\nFurthermore, using $|\\lambda_6|, |\\lambda_7| \\lesssim$ few $\\times 10^{-7}$ to\nsatisfy Eq.~(\\ref{bdL}), and assuming $m_{11}^2$ and $m_{22}^2$ of the order\nof the lightest Higgs mass, it follows from Eqs.~\\eqref{mtherm}\nand~\\eqref{thetatapprox} that\n\\begin{equation}\\label{m12bound}\n|m_{12}^2| \\lesssim (100\\,\\textrm{MeV})^2,\n\\end{equation}\nin the Yukawa eigenbasis.\n\nWe remark that, in obtaining the bound (\\ref{topbd}), we approximate\n$m_t=(|Y^t_1|^2 + |Y^t_2|^2)^{1\/2}\\, v $, that is, we neglect the\nmisalignment between the top Yukawa coupling vector and the zero-temperature\nVEVs. This could underestimate the magnitude of the Yukawa coupling (as arises\nfor the $b$ and $\\tau$ in the large $\\tan \\beta$ limit of the supersymmetric\nSM). Therefore, the interaction rates we obtain will be lower bounds. Similar\nbounds apply to other fermions $f$, with the replacement $m_t \\to m_f$ (and\n$\\alpha_s \\to \\alpha_{\\rm QED}\/4 $ for leptons). This leads to the bounds\n\\begin{eqnarray}\\label{thetabtau}\n\\theta_b &\\lesssim& 10^{-5} \\to 10^{-7}\\quad \\text{for the b quark}, \\nonumber\\\\\n\\theta_\\tau &\\lesssim& 10^{-4} \\to 10^{-6}\\quad \\text{for the $\\tau$-charged lepton}.\n\\end{eqnarray}\nThe weaker limit corresponds to $|Y^f_1|^2 +|Y^f_2|^2=(m_f\/v)^2$ and the\nstronger one to $|Y^f_1|^2 +|Y^f_2|^2 \\sim 1$.\n\n\\subsection{Basis-independent conditions}\n\\label{sec:basis}\n\nIn this section, the conditions given in Eqs.~(\\ref{bdL}) and (\\ref{topbd}),\nwhich ensure the survival of a relative Higgs asymmetry, are expressed in a\nway which is independent of the (Higgs) basis transformation\n\\begin{equation}\n\\Phi \\rightarrow \\mathbf{U}\\, \\Phi,\n\\label{HBT}\n\\end{equation}\nwhere $\\Phi = ( \\phi_1,\\ \\phi_2)^T$ and $\\mathbf{U}$ is a $2 \\times 2$ unitary\nmatrix in Higgs flavour space. In Refs.~\\cite{Lavoura:1994fv,DH},\nbasis-independent combinations of potential parameters were constructed by\ncontracting the parameters with the Higgs VEVs (a vector in Higgs doublet\nspace). We will construct similar invariants here, but replacing the Higgs VEV\nwith the top Yukawa coupling, which is more relevant for our scenario, and is\nalso a vector in Higgs space (in the one generation approximation). Indeed,\none can combine the top Yukawa couplings in\n\\begin{equation}\n- {\\cal L}_Y\n=\n\\overline{t_L} \\left( y_1^t,\\ y_2^t\\right)\\,\n\\left(\n\\begin{array}{cc}\n\\phi_1\\\\\n\\phi_2\n\\end{array}\n\\right)\\, t_R\n+ \\textrm{h.c.},\n\\end{equation}\ninto a vector\n\\begin{equation}\n\\hat{y}^t\n=\n\\frac{1}{\\sqrt{|y^t_1|^2 +|y^t_2|^2}}\n\\left(\n\\begin{array}{c}\ny^t_1\\\\\ny^t_2\n\\end{array}\n\\right),\n\\label{hat_y}\n\\end{equation}\ntransforming as $\\hat{y}^t \\rightarrow \\mathbf{U}\\, \\hat{y}^t$,\nand its orthogonal\n\\begin{equation}\n\\hat{\\epsilon}^t\n=\n\\frac{1}{\\sqrt{|y^t_1|^2 +|y^t_2|^2}}\n\\left(\n\\begin{array}{c}\n- y^{t\\,\\ast}_2\\\\\ny^{t\\,\\ast}_1\n\\end{array}\n\\right),\n\\label{hat_eps}\n\\end{equation}\ntransforming as $\\hat{\\epsilon}^t \\rightarrow\n[\\textrm{det}\\,\\mathbf{U}]^{-1}\\, \\mathbf{U}\\, \\hat{\\epsilon}^t$. In the top\nbasis (see Appendix~\\ref{sec:appendix}), these vectors become\n\\begin{equation}\n\\hat{y}^t\n=\n\\left(\n\\begin{array}{c}\n1\\\\\n0\n\\end{array}\n\\right),\n\\ \\ \\ \\\n\\hat{\\epsilon}^t\n=\n\\left(\n\\begin{array}{c}\n0\\\\\n1\n\\end{array}\n\\right).\n\\label{topbasis}\n\\end{equation}\nFrom Eq.~(\\ref{topbd}), it is clear that the direction in Higgs space of the\ntop Yukawa coupling $\\hat{y}^t$ should approximately correspond to $\\phi_1$\nor $\\phi_2$ of the thermal mass eigenstate basis. We then simply impose the\nbounds of Eqs.~(\\ref{bdL}) and (\\ref{topbd}) in the basis of\nEq.~(\\ref{topbasis}).\n\nIt is convenient to introduce some notation patterned on Ref.~\\cite{DH}. The\nquartic Higgs interactions can be represented via a four-index tensor which\nappears in the Lagrangian as $\\frac{1}{2} Z_{a \\bar{b} c \\bar{d}}\n\\Phi_{\\bar{a}}^\\dagger \\Phi_b \\Phi_{\\bar{c}}^\\dagger \\Phi_d$ where\n$\\Phi^\\dagger = (\\phi_1^\\dagger, \\phi_2^\\dagger)$, and $a,b,c,d=1,2$. The\nbarred (unbarred) notation keeps track of which indices transform as\n$\\mathbf{U}^\\dagger$ ($\\mathbf{U}$), under the basis\ntransformation~\\eqref{HBT}. The elements of $Z_{a \\bar{b} c \\bar{d}}$ are\n\\begin{eqnarray}\n&& Z_{1\\bar{1}1\\bar{1}}=\\lambda_1\\,,\\qquad\\qquad \\,\\,\\phantom{Z_{2\\bar{2}2\\bar{2}}=}\nZ_{2\\bar{2}2\\bar{2}}=\\lambda_2\\,,\\nonumber\\\\\n&& Z_{1\\bar{1}2\\bar{2}}=Z_{2\\bar{2}1\\bar{1}}=\\lambda_3\\,,\\qquad\\qquad\nZ_{1\\bar{2}2\\bar{1}}=Z_{2\\bar{1}1\\bar{2}}=\\lambda_4\\,,\\nonumber \\\\\n&& Z_{1\\bar{2}1\\bar{2}}=\\lambda_5\\,,\\qquad\\qquad \\,\\,\\phantom{Z_{2\\bar{2}2\\bar{2}}=}\nZ_{2\\bar{1}2\\bar{1}}=\\lambda_5^*\\,,\\\\\n&& Z_{1\\bar{1}1\\bar{2}}=Z_{1\\bar{2}1\\bar{1}}=\\lambda_6\\,,\\qquad\\qquad\nZ_{1\\bar{1}2\\bar{1}}=Z_{2\\bar{1}1\\bar{1}}=\\lambda_6^*\\,,\\nonumber \\\\\n&& Z_{2\\bar{2}1\\bar{2}}=Z_{1\\bar{2}2\\bar{2}}=\\lambda_7\\,,\\qquad\\qquad\nZ_{2\\bar{2}2\\bar{1}}=Z_{2\\bar{1}2\\bar{2}}=\\lambda_7^*\\,.\\nonumber\n\\label{znum}\n\\end{eqnarray}\nBy analogy with the invariants $|Z_5|$, $|Z_6|$, and $|Z_7|$ presented in\nRef.~\\cite{DH}, the following basis invariant quantities can be constructed:\n\\begin{eqnarray}\n|S_5|\n& \\equiv &\n| Z_{a \\bar{b} c \\bar{d}}\n\\, \\hat{y}_{\\bar{a}}^{t \\ast}\n\\, \\hat{\\epsilon}_{b}^{t}\n\\, \\hat{y}_{\\bar{c}}^{t \\ast}\n\\, \\hat{\\epsilon}_{d}^{t} |,\n\\label{szvv5}\n\\nonumber\\\\\n|S_6|\n& \\equiv &\n| Z_{a \\bar{b} c \\bar{d}}\n\\, \\hat{y}_{\\bar{a}}^{t \\ast}\n\\, \\hat{y}_{b}^{t}\n\\, \\hat{y}_{\\bar{c}}^{t \\ast}\n\\, \\hat{\\epsilon}_{d}^{t} |,\n\\label{szvv6}\\\\\n|S_7|\n& \\equiv &\n| Z_{a \\bar{b} c \\bar{d}}\n\\, \\hat{y}_{\\bar{a}}^{t \\ast}\n\\, \\hat{\\epsilon}_{b}^{t}\n\\, \\hat{\\epsilon}_{\\bar{c}}^{t \\ast}\n\\, \\hat{\\epsilon}_{d}^{t} |.\\nonumber\n\\label{szvv7}\n\\end{eqnarray}\nThese correspond to $|\\lambda_5|$, $|\\lambda_6|$, and $|\\lambda_7|$ in the\nbasis of Eq. (\\ref{topbasis}) and, consequently, $|S_n| \\lesssim {\\rm few}\n\\times 10^{-7}$ to satisfy Eq.~(\\ref{bdL}).\n\nAs seen in Appendix~\\ref{sec:appendix}, the rotation angle between the thermal\nmass eigenstate basis and the top Yukawa eigenbasis of Eq.~(\\ref{topbasis})\ncan be written in the top basis as\n\\begin{equation}\n|\\tan{(2 \\theta_{t})}|\n=\n\\frac{2 |m_{12}^{t 2}|}{|m_{11}^{t 2} - m_{22}^{t 2}|}\n=\n\\frac{2\n\\left|m_{a \\bar{b}}^{t 2}\\ \\hat{y}_{\\bar{a}}^{t \\ast}\\ \\hat{\\epsilon}_b^t\n\\right| }{ \\left|\nm_{a \\bar{b}}^{t 2}\\ \\hat{y}_{\\bar{a}}^{t \\ast}\\ \\hat{y}_b^t -\nm_{a \\bar{b}}^{t 2}\\ \\hat{\\epsilon}_{\\bar{a}}^{t \\ast}\\ \\hat{y}_b^t\n\\right|},\n\\end{equation}\nwhere the last expression is manifestly basis invariant and, according to\nEq.~\\eqref{thetatbound}, $\\theta_t \\lesssim {\\rm few} \\times 10^{-7}$ to\nsatisfy Eq.~(\\ref{topbd}).\n\nFinally, the misalignment angles between the top, bottom and tau eigenbases\ncan be formulated in basis-independent notation as\n\\begin{eqnarray}\n{\\rm min} {\\Big\\{} \\hat{y}^b\\cdot \\hat{y}^t,\n \\hat{y}^b\\cdot \\hat{\\epsilon}^t {\\Big\\}} &\\simeq & \\theta_b \\lesssim 10^{-5}\n\\to 10^{-7},\n\\nonumber\\\\\n{\\rm min} {\\Big\\{} \\hat{y}^\\tau\\cdot \\hat{y}^t,\n \\hat{y}^\\tau\\cdot \\hat{\\epsilon}^t {\\Big\\}} &\\simeq & \\theta_\\tau\n \\lesssim 10^{-4}\\to 10^{-6},\n\\end{eqnarray}\nwhere $\\hat{y}^b$ and $\\hat{y}^\\tau$ are defined analogously to $\\hat{y}^t$\nin Eq.~\\eqref{hat_y} and the upper bounds on the right-hand sides follow from\nEq.~\\eqref{thetabtau}.\n\nIn summary, in a 2HDM prior to the EWPT, a relative Higgs asymmetry can\nsurvive provided the Yukawa interactions and the Higgs potential have a\ncertain form. The Higgs potential parameters should satisfy the constraints\n$|\\lambda_n| \\lesssim$ few $\\times 10^{-7}\\ (n=5,6,7)$, and $|m_{12}^2|\n\\lesssim $ (100 MeV)$^2$. In the same basis, SM singlet fermions of a given\ncharge (up-type quarks, down-type quarks, charged leptons) should interact\nwith approximately only one Higgs field, that is, the model should be of type\nI, II, X, or Y.\n\n\\section{Chemical equilibrium relations}\n\\label{sec:chemeq}\n\nLet us now study the redistribution of asymmetries in conserved quantum\nnumbers due to interactions in equilibrium. We neglect lepton flavour\nasymmetries, so that the (exactly and effectively) conserved quantum numbers\nare the hypercharge, $B-L$, and the relative Higgs asymmetry. Assuming that\nthe asymmetries in all species are small, they are related to the chemical\npotential $\\mu$ as\n\\begin{equation}\nn_i - n_{\\overline{i}} =\n\\frac{g_i T^2}{6} \\, \\mu_i \\times\n\\left\\{\n\\begin{array}{cl}\n2 & \\textrm{for bosons}\\\\\n1 & \\textrm{for fermions}\n\\end{array}\n\\right.,\n\\end{equation}\nwhere $g_i$ is the number of degrees of freedom of the particle.\n\nWe take the thermal bath to contain the SM fermions and gauge bosons, and two\nHiggs doublets. We consider temperatures just prior to the EWPT, when all the\nYukawa interactions are in equilibrium, but gauge symmetries are unbroken, so\nthat the gauge bosons have zero chemical potential. Our aim is to investigate\nwhether a Higgs asymmetry, as given in Eq.~\\eqref{asym}, can be used to\ngenerate a baryon asymmetry.\n\nIf an interaction is in chemical equilibrium, then the sum of the chemical\npotentials of the participating particles should vanish. The SM Yukawa\ninteractions impose the relations\n\\begin{eqnarray}\n- \\mu_q + \\mu_{\\phi_d} + \\mu_{d_R} &=& 0,\n\\label{yuk_d}\n\\\\\n- \\mu_q - \\mu_{\\phi_u} + \\mu_{u_R} &=& 0,\n\\label{yuk_u}\n\\\\\n- \\mu_\\ell + \\mu_{\\phi_e} + \\mu_{e_R} &=& 0,\n\\label{yuk_e}\n\\end{eqnarray}\nwhere $\\phi_d$, $\\phi_u$ and $\\phi_e$ denote the scalar that couples to the\ndown-type quarks, up-type quarks and charged leptons, respectively. Since in\nthe usual notation $\\phi_u = \\phi_2$, the various models in Table~\\ref{table1}\ndiffer by whether $\\phi_d$ and\/or $\\phi_e$ coincide with $\\phi_2$.\n\nThe electroweak sphalerons impose\n\\begin{equation}\n3 \\mu_q + \\mu_\\ell = 0,\n\\label{Ewk_sphal}\n\\end{equation}\nwhile the QCD sphalerons lead to the chemical equilibrium condition\n\\begin{equation}\n2 \\mu_q - \\mu_{u_R} - \\mu_{d_R} = 0.\n\\label{QCD_sphal}\n\\end{equation}\nAdding Eqs.~\\eqref{yuk_d}, \\eqref{yuk_u}, and \\eqref{QCD_sphal}, we find\n\\begin{equation}\n\\mu_{\\phi_d} - \\mu_{\\phi_u} = 0.\n\\label{crucial}\n\\end{equation}\nIn type II and type Y models, where $\\phi_2$ couples to up-type quarks, and\n$\\phi_1$ to down-type quarks, this forces $\\mu_{\\phi_1} - \\mu_{\\phi_2}$ to\nvanish. Therefore, in 2HDM of type II and type Y, a relative Higgs asymmetry\nwould be washed out. In contrast, in type I and type X models, $\\phi_d =\n\\phi_2 = \\phi_u$, Eq.~\\eqref{crucial} is trivially satisfied, and thus\none can have $\\mu_{\\phi_1} - \\mu_{\\phi_2} \\neq 0$. Next we show that, provided\na Higgs asymmetry was created in early Universe, it can be used to generate a\nbaryon asymmetry at later times.\n\nThe baryon and lepton number comoving asymmetries are given by\n\\begin{eqnarray}\n\\mathcal{Y}_{\\Delta B} &=&N_g (\n2 \\mu_q + \\mu_{u_R} + \\mu_{d_R})\\,\\frac{T^2}{3s}= 4N_g \\mu_q \\frac{T^2}{3s},\n\\nonumber\\\\\n\\mathcal{Y}_{\\Delta L} &=& N_g (2 \\mu_\\ell + \\mu_{e_R})\\,\\frac{T^2}{3s} =\n- N_g (9 \\mu_q + \\mu_{\\phi_e})\\,\\frac{T^2}{3s},\n\\end{eqnarray}\nwhere $N_g = 3$ is the number of generations, and\nEqs.~\\eqref{yuk_e}-\\eqref{QCD_sphal} have been used to rewrite the right-hand\nsides of these expressions. As a result,\n\\begin{equation}\n\\mathcal{Y}_{\\Delta B} - \\mathcal{Y}_{\\Delta L} =\nN_g \\left( 13\\, \\mu_q + \\mu_{\\phi_e} \\right)\\,\\frac{T^2}{3s}.\n\\label{BmL}\n\\end{equation}\n\nFinally, hypercharge (or, equivalently, electric) neutrality of the plasma\ngives\n\\begin{eqnarray}\nN_g &&\\left(\n- \\mu_{e_R} - \\mu_\\ell + \\mu_q + 2 \\mu_{u_R} - \\mu_{d_R}\n\\right)\n+ 2 ( \\mu_{\\phi_1} + \\mu_{\\phi_2} )\n\\nonumber\\\\\n=&& N_g \\left(\n8 \\mu_q + \\mu_{\\phi_e} + 3 \\mu_{\\phi_2}\n\\right)\n+ 2 ( \\mu_{\\phi_1} + \\mu_{\\phi_2} )=0.\n\\label{Qem}\n\\end{eqnarray}\nThe equilibrium baryon asymmetry can then be written as a function of $B-L$\nand the Higgs asymmetry:\n\\begin{eqnarray}\n\\mathcal{Y}_{\\Delta B}\n&=&\n\\frac{8}{23} (\\mathcal{Y}_{\\Delta B} - \\mathcal{Y}_{\\Delta L})\n+ \\frac{3}{46}\n\\left( \\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\right)\n\\hspace{5mm}\n\\textrm{for type I},\n\\label{BI_TypeI}\n\\\\\n\\mathcal{Y}_{\\Delta B}\n&=&\n\\frac{8}{23} (\\mathcal{Y}_{\\Delta B} - \\mathcal{Y}_{\\Delta L})\n- \\frac{33}{92}\n\\left( \\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\right)\n\\hspace{5mm}\n\\textrm{for type X}.\n\\label{BI_TypeX}\n\\end{eqnarray}\n\nEqs.~\\eqref{BI_TypeI} and \\eqref{BI_TypeX} give a baryon asymmetry in the\npresence of a relative Higgs asymmetry, even if $\\mathcal{Y}_{\\Delta B}\n-\\mathcal{Y}_{\\Delta L}=0$. Thus, in these cases, the baryon asymmetry is due\nexclusively to the initial imbalance between the asymmetry in $\\phi_1$ and\nthe asymmetry in $\\phi_2$. We dub this scenario as split Higgsogenesis. As\nfar as we know, this is a novel mechanism for baryogenesis. This is\nreminiscent of certain asymmetric dark matter (DM) models~\\cite{ADMrev},\nwhere an asymmetry is generated in a new dark sector (which contains the DM\ncandidate), and then is shared with the SM fermions. The role that the Higgs\ncan play in transferring asymmetries between the SM fermions and the dark\nsector has been recently emphasized in Ref.~\\cite{ST}. However, $\\phi_1$ does\nnot seem to be a successful asymmetric DM candidate in the simple model\ndiscussed here.\\footnote{Symmetric dark matter and electroweak baryogenesis\nhave been recently discussed in the inert doublet model in\nRefs.~\\cite{BAUIDM}.}\n\nLet us assume that, indeed, $\\mathcal{Y}_{\\Delta B} =\\mathcal{Y}_{\\Delta L}$. Using\nEqs.~\\eqref{BmL}-\\eqref{BI_TypeX}, we find\n\\begin{eqnarray}\n\\mathcal{Y}_{\\Delta B}\n&=&\n\\frac{3}{46}\n\\left( \\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\right)\n=\n-\\frac{6}{13}\\, \\mathcal{Y}_{\\Delta \\phi_2} = \\frac{6}{79}\\, \\mathcal{Y}_{\\Delta \\phi_1}\n\\hspace{6mm}\n\\textrm{for type I},\n\\label{B_TypeI}\n\\\\*[2mm]\n\\mathcal{Y}_{\\Delta B}\n&=&\n- \\frac{33}{92}\n\\left( \\mathcal{Y}_{\\Delta \\phi_1} - \\mathcal{Y}_{\\Delta \\phi_2} \\right)\n=\n-\\frac{6}{13}\\, \\mathcal{Y}_{\\Delta \\phi_1} = \\frac{66}{41}\\, \\mathcal{Y}_{\\Delta \\phi_2}\n\\hspace{5mm}\n\\textrm{for type X}.\n\\label{B_TypeX}\n\\end{eqnarray}\nThe type I model is particularly interesting because $\\phi_1$ does not couple\nto any fermion and could act as dark matter. In that case,\n$\\mathcal{Y}_{\\Delta \\phi_1}\/\\mathcal{Y}_{\\Delta B} = 79\/6$, so the DM scalar\nshould be lighter than the proton\\footnote{ Alternatively, one could assume a\nprimordial $B-L$ asymmetry, no relative Higgs asymmetry, and let the\n$\\lambda_{5}$ coupling (allowed by the $Z_2$ symmetry which ensures DM\nstability) to equilibrate the asymmetry between the two Higgs fields. Then,\nEq.~\\eqref{Qem} yields a Higgs asymmetry smaller than the baryon asymmetry: $\n\\mathcal{Y}_{\\Delta \\phi_1} = \\mathcal{Y}_{\\Delta \\phi_2} = -\n\\mathcal{Y}_{\\Delta B}\/8$. This corresponds to a scalar DM mass $\\sim 20$\nGeV, which is ruled out by the width of the $Z$ boson. Furthermore,\n$\\lambda_5$ would mediate DM-anti-DM oscillations which would wash out the\nasymmetry.} to obtain $\\Omega_{\\textrm{DM}} \\sim 5\\, \\Omega_{B}$. We recall\nthat the mass density of baryons in the Universe today, as inferred from WMAP\nin the context of $\\Lambda$CDM cosmology, is~\\cite{Komatsu:2010fb}\n\\begin{equation}\\label{nBobs}\n \\frac{m_p}{\\rho_c}( n_B - n_{\\overline B})=\n \\Omega_{B} h^2 = 0.02255 \\pm 0.0054,\n\\end{equation}\nor equivalently,\n\\begin{equation}\\label{YBobs}\n \\mathcal{Y}_{\\Delta B} = (8.79\\pm0.44)\\times 10^{-11},\n\\end{equation}\nwhere $m_p$ is the proton mass, $h \\equiv H_0\/(100\\, $km s$^{-1}\\,$Mpc$^{-1})\n= 0.742 \\pm 0.036$ is the present Hubble parameter, and $\\rho_c = 3H_0^2\/(8\n\\pi G)$ is the critical density of a spatially flat Universe. On the other\nhand, for the $\\Lambda$CDM cosmology with three light neutrinos, the cold dark\nmatter relic abundance is $\\Omega_{\\textrm{DM}} h^2 = 0.1126\\pm\n0.0036$~\\cite{Komatsu:2010fb}, so that the ratio of dark matter particles to\nbaryons is $\\mathcal{Y}_{\\textrm{DM}}\/\\mathcal{Y}_{\\Delta B} \\sim 5\\,\nm_p\/m_{\\textrm{DM}}$.\n\n\\section{Simple split-Higgsogenesis scenarios}\n\\label{sec:scenarios}\n\nOur goal in this section is to provide a few simple scenarios for\nbaryogenesis through split-Higgsogenesis, where the cosmological baryon\nasymmetry could in principle be generated via the out-of-equilibrium decay of\nheavy singlet scalars into Higgs doublets.\n\n\\subsection{One extra singlet scalar}\n\nWe consider an inert 2HDM extended by one real scalar singlet. The two Higgs\ndoublets, $\\phi_1$ and $\\phi_2$, and the singlet scalar $S$ transform under\n$Z_2$ as\n\\begin{equation}\n\\phi_1 \\rightarrow - \\phi_1,\\quad\n\\phi_2 \\rightarrow \\phi_2,\\quad\nS \\rightarrow -S.\n\\label{Z2_2d1s}\n\\end{equation}\nThe $Z_2$-invariant Higgs potential can be written as\n\\begin{equation}\nV = V_\\phi + V_S + V_{S \\phi},\n\\label{VPPS}\n\\end{equation}\nwhere\n\\begin{eqnarray}\\label{V_vp_Z2}\nV_\\phi\n&=&\nm_{11}^2\\, |\\phi_1|^2 + m_{22}^2\\, |\\phi_2|^2\n+\\, \\tfrac{1}{2}\\, \\lambda_1\\, |\\phi_1|^4 + \\tfrac{1}{2}\\, \\lambda_2\\,\n|\\phi_2|^4 + \\lambda_3\\, |\\phi_1|^2 |\\phi_2|^2 + \\lambda_4\\, (\\phi_1^\\dagger\n\\phi_2)(\\phi_2^\\dagger \\phi_1)\n\\nonumber\\\\\n&&\n+\\, \\tfrac{1}{2}\\, \\lambda_5 \\left[ (\\phi_1^\\dagger \\phi_2)^2 +\n\\textrm{h.c.} \\right],\\nonumber\\\\\nV_S &=& M^2\\, S^2 + \\lambda_S\\, S^4,\\\\\nV_{S \\phi}\n&=&\nz_1\\,M (\\phi_1^\\dagger \\phi_2)\\, S + z_1^\\ast\\,M (\\phi_2^\\dagger \\phi_1)\\, S\n+\n\\beta_1\\, |\\phi_1|^2\\, S^2 + \\beta_2\\, |\\phi_2|^2\\, S^2.\\nonumber\n\\end{eqnarray}\nAll parameters are real but $z_1$, so that CP violation in the scalar sector\nis related with a complex $z_1$.\n\nWe address now the question whether this model can be used to generate a CP\nasymmetry in the Higgs sector that could be converted into a baryon\nasymmetry. The basic idea is analogous to that of the standard leptogenesis\nscenario. A population of $S$'s is produced through scattering processes at\ntemperatures $T \\sim M \\gg m_{\\phi_1,\\phi_2}$. This population decays away at\n$T < M$, when the singlet scalar equilibrium density is Boltzmann suppressed.\nIf the interactions of the heavy singlet $S$ are CP-violating, and provided\nthat the relevant interactions are out of equilibrium, a net Higgs asymmetry\ncan be generated. The latter is then converted into a baryon asymmetry by the\nsphalerons. To illustrate our mechanism, let us consider the tree-level and\none-loop diagrams\\footnote{There is also a one-loop vertex correction to the\ntree-level diagram, but since it carries the same phase $z_1$, it does not\ncontribute to the Higgs CP asymmetry.} depicted in Fig.~\\ref{fig1}.\n\n\\begin{figure}[t]\n\\includegraphics*[height=3.5cm]{fig1.eps}\n\\caption{\\label{fig1}Diagrams contributing to the Higgs CP asymmetry. The\nnotation $1$ and $\\bar{2}$ refers to $\\phi_1$ and $\\bar{\\phi}_2$,\nrespectively.}\n\\end{figure}\nWe have two contributions with different CP-odd phases; $z_1$ and $z_1^\\ast$.\nBecause the second diagram is a loop diagram, a cut on this diagram leads to\nan absorptive part that contains the CP-even phase needed for CP violation in\ndecays. As a result, the interference of the tree-level and one-loop\namplitudes leads to a nonvanishing CP asymmetry in the final Higgs states.\nDefining this asymmetry as\n\\begin{equation} \\label{CPasym}\n\\epsilon = \\frac{\\Gamma(S\\rightarrow \\phi_1\\,\\bar{\\phi}_2)-\\Gamma(S\\rightarrow\n\\bar{\\phi}_1\\,\\phi_2)}{ \\Gamma(S\\rightarrow\n\\phi_1\\,\\bar{\\phi}_2)+\\Gamma(S\\rightarrow \\bar{\\phi}_1\\,\\phi_2)},\n\\end{equation}\nwe find\n\\begin{equation}\n\\epsilon \\simeq\\frac{\\overline{\\left|c_0 \\mathcal{A}_0+c_1\n\\mathcal{A}_1\\right|^2} -\\overline{\\left|c^\\ast_0 \\mathcal{A}_0+c^\\ast_1\n\\mathcal{A}_1\\right|^2}}{ \\overline{\\left|c_0 \\mathcal{A}_0+c_1\n\\mathcal{A}_1\\right|^2}+ \\overline{\\left|c^\\ast_0 \\mathcal{A}_0+c^\\ast_1\n\\mathcal{A}_1\\right|^2}} \\simeq -4\\frac{\\text{Im}\\left[c_0^\\ast c_1\\right]\n\\text{Im}\\left[\\overline{\\mathcal{A}_0^\\ast \\mathcal{A}_1}\\right]}{\n2|c_0|^2\\overline{|\\mathcal{A}_0|^2}},\n\\end{equation}\nwhere $\\mathcal{A}_0$ and $\\mathcal{A}_1$ are the tree-level and one-loop\namplitudes, respectively. For the decay of Fig.~\\ref{fig1}, one has $c_0 = -\nz_1$, $c_1= 3\\,\\lambda_5\\, z_1^\\ast$ and $\\mathcal{A}_0=1$. Thus, the weak\nphase gives\n\\begin{equation}\n\\text{Im}\\left[c_0^\\ast\\, c_1\\right]=- 3\\, \\text{Im}\\left[z_1^{\\ast}\\,\n\\lambda_5\\, z_1^{\\ast} \\right],\n\\end{equation}\nwhile the strong phase comes from\n\\begin{equation}\n\\text{Im}\\left[\\overline{\\mathcal{A}_0^\\ast \\mathcal{A}_1}\\right]\n=-\\frac{1}{16\\pi}.\n\\end{equation}\nWe then get\n\\begin{equation} \\label{eps1}\n\\epsilon \\simeq - \\frac{3}{8\\pi} \\frac{\\text{Im}\\left[z_1^{\\ast}\\, \\lambda_5\\, z_1^{\\ast} \\right]}{2 |z_1|^2} =\n\\frac{3}{16\\pi} \\lambda_5 \\sin[2\\,\\text{arg}(z_1)].\n\\end{equation}\nThus, in this simple scenario, the Higgs asymmetry is controlled by the\nstrength of the quartic parameter $\\lambda_5$ and the phase of the coupling\n$z_1$.\n\nThe final baryon asymmetry (baryon-to-entropy ratio) can be approximated as\n\\begin{equation}\\label{YB}\n \\mathcal{Y}_{\\Delta B} \\simeq \\mathcal{Y}^{\\rm eq}_S\\times C\\times\\epsilon\\,\n \\eta,\n\\end{equation}\nwhere the first factor is the equilibrium $S$ number density divided by the\nentropy density, the second factor is the fraction of the Higgs asymmetry\nconverted into a baryon asymmetry by the sphalerons ($C=3\/46$ in the present\ncase), and the efficiency factor $\\eta\\, (0 \\leq \\eta \\leq 1)$ measures how\nefficient the out-of-equilibrium $S$-decays are in producing the asymmetry.\n\nAlthough a precise computation of $\\eta$ requires the solution of a full set\nof Boltzmann equations, simple analytical estimates can be given. It is useful\nto introduce the decay parameter\n\\begin{equation}\\label{Kappa}\n K=\\frac{\\Gamma_D(T=0)}{H(T=M)},\n\\end{equation}\nwhere $\\Gamma_D=|z_1|^2 M\/(8\\pi)$ is the tree-level decay rate of the singlet\n$S$ into the two Higgs doublets. In the so-called weak washout regime ($K \\ll\n1$), i.e. when the scalar singlet decays strongly out of equilibrium, the\nefficiency factor is $\\eta\\simeq 1$. In the strong washout regime ($K\\gg 1$),\nthe efficiency does not depend on the initial conditions, and is mildly\nsuppressed as $\\eta \\simeq 1\/K$. For intermediate values of $K$ ($K \\lesssim\n1$ or $K \\gtrsim 1$) the efficiency depends on the assumed initial conditions.\nWe can roughly approximate it as $\\eta \\sim \\min(1, 1\/K)$, if $S$ has thermal\ninitial abundance, or $\\eta \\sim \\min(K, 1\/K)$, if $S$ has zero initial\nabundance.\n\nAn estimate for $\\mathcal{Y}_{\\Delta B}$ can be obtained from\nEq.~\\eqref{YB} in the form:\n\\begin{equation}\\label{YBapp}\n \\mathcal{Y}_{\\Delta B} \\simeq \\frac{135 \\zeta(3)}{92\\pi^4 g_\\ast}\n \\epsilon\\, \\eta \\simeq 2\\times10^{-4} \\epsilon\\, \\eta .\n\\end{equation}\nThis is to be compared with the WMAP inferred value given in\nEq.~\\eqref{YBobs}. Since Eq.~\\eqref{eps1} leads to the upper bound $|\\epsilon|\n\\lesssim 6\\times 10^{-2} \\lambda_5$, then Eq.~\\eqref{YBapp}, when combined with\nEq.~\\eqref{YBobs}, requires\n\\begin{equation}\n\\lambda_5 \\, \\eta \\gtrsim 7\\times 10^{-6}.\n\\end{equation}\n\nNotice that between the mass scale $M$ and the EW scale an effective\nquartic coupling $\\lambda_5^\\text{eff}=\\lambda_5 +\\tfrac{1}{2} z_1^2$ is\ngenerated by $S$-exchange. Recalling that, for the relative asymmetry between\n$\\phi_1$ and $\\phi_2$ to survive, interactions which exchange $\\phi_1\n\\leftrightarrow \\phi_2$ must be out of equilibrium until the electroweak\nscale, then Eq.~\\eqref{bdL} imposes $|\\lambda_5^\\text{eff}| \\lesssim {\\rm\nfew}\\times 10^{-7}$. Thus, in this simple setup, unless there is a fine-tuned\ncancellation between $\\lambda_5$ and $z_1^2$ to satisfy this bound, we cannot\naccommodate the observed baryon asymmetry~\\eqref{YBobs}, even with a maximal\nefficiency $\\eta\\simeq 1$.\n\n\\subsection{One extra singlet and a third doublet}\n\nWe consider now a model with three doublet scalars and one real scalar singlet.\nThe Higgs doublets, $\\phi_1$, $\\phi_2$ and $\\phi_3$, and the singlet scalar\n$S$ transform under $Z_2$ as\n\\begin{equation}\n\\phi_1 \\rightarrow - \\phi_1,\\quad\n\\phi_3 \\rightarrow - \\phi_3,\\quad\n\\phi_2 \\rightarrow \\phi_2,\\quad\nS \\rightarrow -S.\n\\label{Z2_3d1s}\n\\end{equation}\nThe singlet and the third doublet $\\phi_3$ will be significantly heavier\nthan the EW scale; $\\phi_1$ and $\\phi_3$ are defined as the $Z_2$-odd mass\neigenstates. The $Z_2$-invariant Higgs potential can be written as\n\\begin{equation}\nV_3 = V + \\Delta V_\\phi + \\Delta V_{S \\phi},\n\\label{V3}\n\\end{equation}\nwhere $V$ is given in Eqs.~\\eqref{VPPS}-\\eqref{V_vp_Z2} and\n\\begin{eqnarray}\\label{V3_vp_Z2}\n\\Delta V_\\phi\n&=&\nm_{33}^2\\, |\\phi_3|^2\n+ \\, \\tfrac{1}{2}\\, \\lambda_{3333}\\, |\\phi_3|^4\n+ \\lambda_{1133}\\, |\\phi_1|^2 |\\phi_3|^2\n+ \\lambda_{2233}\\, |\\phi_2|^2 |\\phi_3|^2\n\\nonumber\\\\\n&&\n + \\lambda_{1331}\\, (\\phi_1^\\dagger \\phi_3)(\\phi_3^\\dagger \\phi_1)\n+ \\lambda_{2332}\\, (\\phi_2^\\dagger \\phi_3)(\\phi_3^\\dagger \\phi_2)\n\\nonumber\\\\\n&&\n+ \\left[\\tfrac{1}{2}\\lambda_{1313}(\\phi_1^\\dagger \\phi_3)^2\n+ \\tfrac{1}{2}\\lambda_{3232}(\\phi_3^\\dagger \\phi_2)^2\n+ \\lambda_{1223}(\\phi_1^\\dagger \\phi_2) (\\phi_2^\\dagger \\phi_3)\n+ \\lambda_{1232}(\\phi_1^\\dagger \\phi_2) (\\phi_3^\\dagger \\phi_2) \\right.\\nonumber\\\\\n&&\n+ \\left. \\lambda_{1311}(\\phi_1^\\dagger \\phi_3) (\\phi_1^\\dagger \\phi_1)\n+ \\lambda_{1322}(\\phi_1^\\dagger \\phi_3) (\\phi_2^\\dagger \\phi_2)\n+ \\lambda_{1333}(\\phi_1^\\dagger \\phi_3) (\\phi_3^\\dagger \\phi_3) + \\textrm{H.c.} \\right]\n ,\\nonumber\\\\\n\\Delta V_{S \\phi}\n&=&\nz_3\\,M (\\phi_3^\\dagger \\phi_2)\\, S + z_3^\\ast\\,M (\\phi_2^\\dagger \\phi_3)\\, S\n+\n\\beta_3\\, |\\phi_3|^2\\, S^2 + \\left[\\beta_{13}\\, \\phi_1^\\dagger \\phi_3 \\, S^2 +\n \\textrm{H.c.} \\right].\n\\end{eqnarray}\n\nThe basic idea is similar to the 2HDM with an extra singlet. A population of\n$S$'s is produced through scattering processes at temperatures $T \\sim M >\nm_{33} \\gg m_{\\phi_1,\\phi_2}$. This population decays away at $T < M$, when\nthe singlet scalar equilibrium density is Boltzmann suppressed. If the\ninteractions of the heavy singlet $S$ are CP-violating, asymmetries among the\nthree Higgs doublets can be generated. When washout interactions are out of\nequilibrium, the asymmetries can survive. The $\\phi_3$ later decay to\n$\\phi_1$, leaving an asymmetry between $\\phi_2$ and $\\phi_1$. Assuming the\nasymmetry from $S\\to \\phi_1 \\phi_2^*$ to be negligible, due to the bounds on\n$\\lambda_5$ and $z_1^2$, we neglect it and focus on a possible asymmetry\nfrom $S\\to \\phi_3 \\phi_2^*$.\n\nWe consider tree-level and one-loop diagrams analogous to those depicted in\nFig.~\\ref{fig1}, with $\\phi_1 \\rightarrow \\phi_3$, $z_1 \\rightarrow z_3$, and\n$\\lambda_5 \\rightarrow \\lambda_{3232}$. The interference of the\ntree-level and one-loop amplitudes leads to a nonvanishing CP asymmetry in\nthe final Higgs states:\n\\begin{equation} \\label{CPasym3}\n\\epsilon = \\frac{\\Gamma(S\\rightarrow \\phi_3\\,\\bar{\\phi}_2)-\\Gamma(S\\rightarrow\n\\bar{\\phi}_3\\,\\phi_2)}{ \\Gamma(S\\rightarrow {\\rm all})}\n\\simeq - \\frac{3}{8\\pi} \\frac{\\text{Im}\\left[z_3^{\\ast}\\,\n\\lambda_{3232}\\, z_3^{\\ast} \\right]}{2 |z_3|^2} = \\frac{3}{16\\pi} \\lambda_{3232}\n\\sin[2\\,\\text{arg}(z_3)],\n\\end{equation}\nwhere the contribution of $z_1$ to the total decay rate has been neglected.\n\nFor the relative asymmetry between $\\phi_3 + \\phi_1$ and $\\phi_2$ to survive,\ninteractions which exchange $\\phi_3 $ with $\\phi_2$ must be out of\nequilibrium until the $\\phi_3$ decay. (Recall that we have already imposed\nthat interactions exchanging $\\phi_1 \\leftrightarrow \\phi_2$ are out of\nthermal equilibrium). By extrapolating Eq.~\\eqref{bdL} to higher\ntemperatures, this imposes\n\\begin{equation}\n|z_3|^2 , |\\lambda_{3232}| < {\\rm few}\\times\n10^{-7} \\sqrt{\\frac {m_{33}}{\\text{TeV}}}\\,,\n\\end{equation}\nassuming that $S$-exchange generates an effective\n$\\lambda_{3232}^\\text{eff}=\\lambda_{3232}+\\tfrac{1}{2} z_3^2$ between the\ntemperature scales $M$ and $m_{33}$, and that $\\phi_3$ decay at $T \\simeq\n10\\, m_{33}$. From Eqs.~\\eqref{YBapp} and \\eqref{YBobs}, a large enough\nasymmetry is obtained for $m_{33} \\gtrsim 10^8$~GeV, provided the washout\neffects are not too strong, $\\eta \\simeq \\mathcal{O}(0.1)$. Since the\nefficiency of Higgsogenesis is controlled by the decay parameter $K$ defined\nin Eq.~\\eqref{Kappa},\n\\begin{equation}\nK= \\frac{|z_3|^2}{136\\pi}\\frac{m_P}{M} \\simeq 3 \\times\n\\left(\\frac{|z_3|^2}{10^{-4}}\\right)\\times\\left(\\frac{10^{12}\\,\\text{GeV}}{M}\\right),\n\\end{equation}\nthis implies $M \\gtrsim 10^{12}$~GeV.\n\n\\subsection{Two extra singlet scalars}\n\nLet us now consider a model with two real scalar singlets $S_i\\ (i=1,2)$, both\ntransforming under $Z_2$ as $S_i \\rightarrow - S_i$. The Higgs potential can\nbe written as in Eqs.~\\eqref{VPPS}-\\eqref{V_vp_Z2}, but in this case $V_S$\nand $V_{S \\phi}$ contain additional terms. In particular, $V_{S \\phi}$\ncontains the cubic terms $z_i\\,M_i\\, (\\phi_1^\\dagger \\phi_2)\\, S_i+\nz_i^\\ast\\,M_i\\, (\\phi_2^\\dagger \\phi_1)\\, S_i\\,$, in addition to six quartic\nterms. Similarly, $V_S$ has several quartic terms and, without loss of\ngenerality, we choose a $\\left\\{ S_1, S_2\\right\\}$ basis where the quadratic\nterms are already diagonalized, and assume $M_1 < M_2$.\n\nIn what follows, we make the following simplifying assumptions: The heavy\nsinglet spectrum is hierarchical, $M_1 \\ll M_2$; there is a thermal production\nof $S_1$ and negligible production of $S_2$. With these assumptions, the\nHiggsogenesis mechanism will proceed via the out-of-equilibrium decays of\n$S_1$. The decay $S_1 \\rightarrow \\phi_1 \\bar{\\phi}_2$ is still mediated by\nthe diagrams in Fig~\\ref{fig1}, but we also have the additional (vertex and\nself-energy) diagrams depicted in Fig.~\\ref{fig2}.\n\\begin{figure}[t]\n\\includegraphics*[height=3.5cm]{fig2.eps}\n\\caption{\\label{fig2}New diagrams contributing to the Higgs CP asymmetry in\nthe presence of two singlet scalars. The notation $1$ and $\\bar{2}$ refers to\n$\\phi_1$ and $\\bar{\\phi}_2$, respectively.}\n\\end{figure}\n\nThese diagrams carry the phase of $z_1^\\ast z_2^2$, which will be beaten\nagainst the $z_1$ phase from the tree level diagram in Fig.~\\ref{fig1}. As a\nresult, in this model, even if $\\lambda_5$ vanishes, there is a CP-violating\ncontribution proportional to $\\textrm{Im}(z_1^{\\ast 2} z_2^2)$. The resulting\nCP asymmetry, as given in Eq.~\\eqref{CPasym}, can be evaluated following\nthe standard procedure. Neglecting the $\\lambda_5$ contribution coming from\nthe one-loop diagram of Fig.~\\ref{fig1}, we obtain\n\\begin{equation}\n\\epsilon_1 \\simeq -\\frac{1}{4\\pi}\n\\frac{\\text{Im}\\left[(z^\\ast_1z_2)^2\\right]}{|z_1|^2}\\left[\\frac{1}{2}f(x_2)+g(x_2)\\right],\n\\end{equation}\nwhere $x_2 = M_2^2\/M_1^2$, and\n\\begin{equation}\nf(x)=x\\ln \\Big(\\frac{x}{1+x}\\Big),\\quad g(x)=\\frac{x}{1-x},\n\\end{equation}\nare the vertex and self-energy one-loop functions, respectively.\n\nIn the hierarchical limit, $x_2\\gg 1$, $f(x_2)\\simeq -1$, $g(x_2) \\simeq -1$,\nand the CP asymmetry is approximately given by\n\\begin{equation} \\label{eps2}\n\\epsilon_1 \\simeq \\frac{3}{8\\pi}\n\\frac{\\text{Im}\\left[(z^\\ast_1z_2)^2\\right]}{|z_1|^2}=\\frac{3}{8\\pi} |z_2|^2\n\\sin[2\\arg (z_1^\\ast z_2)].\n\\end{equation}\n\nFrom Eqs.~\\eqref{YBapp}, \\eqref{YBobs} and \\eqref{eps2} we then conclude that\na successful generation of the baryon asymmetry within the present model\nrequires\n\\begin{equation}\\label{z2constraint}\n |z_2|^2\\, \\eta \\gtrsim {\\rm few} \\times 10^{-6},\n\\end{equation}\nwhich in turn implies that $|z_2|^2 \\gtrsim {\\rm few} \\times 10^{-6}$. Yet,\nas in the case with one extra singlet, an effective quartic coupling\n$\\lambda_5^{\\rm eff}=\\lambda_5 + \\tfrac{1}{2} z_1^2 +\\tfrac{1}{2} z_2^2$ is\ngenerated by $S$-exchange between the $M$ and EW temperature scales. The\nlatter should satisfy the bound $|\\lambda_5^\\text{eff}| \\lesssim {\\rm\nfew}\\times 10^{-7}$. So, in this case, to accommodate the observed baryon\nasymmetry~\\eqref{YBobs} we should have some relation between $\\lambda_5$,\n$z_1^2$ and\/or $z_2^2$; for example, $z_1 \\simeq i z_2$ to avoid the\nrestrictive bounds on these couplings.\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this work, we have studied the possibility of generating the cosmological\nbaryon asymmetry in the context of 2HDM extensions of the SM, prior to the\nelectroweak phase transition. We have shown that if the Higgs-flavour\nexchanging interactions are sufficiently slow in the early Universe, then a\nrelative asymmetry among the Higgs doublets corresponds to an effectively\nconserved quantum number. Such a relative Higgs asymmetry can be transformed\ninto a baryon asymmetry by the sphalerons, without the need for $B-L$\nviolation.\n\nAmong the four possible types of $Z_2$ models considered, we have\ndemonstrated that this ``split Higgsogenesis'' mechanism is only possible in\nthe framework of a type-I or type-X 2HDM. We then presented simple scenarios\nto generate a Higgs asymmetry, based on inert type-I 2HDMs extended by heavy\nsinglet scalar fields and\/or one extra Higgs doublet. In the presence of\nCP-violating interactions, the out-of-equilibrium decays of the heavy\nsinglets into the Higgs doublets can produce a net Higgs asymmetry and the\nmechanism of baryogenesis through (split) Higgsogenesis can be viable. Since\na successful implementation of our mechanism requires the scalar potential\nparameters to satisfy definite bounds, we have also paid particular attention\nto their basis-independent formulation.\n\n\\acknowledgments\n We are grateful to A. Barroso, M.~Krawczyk and R. Santos for\nuseful discussions. S.D. acknowledges partial support from the EU FP7 ITN\nINVISIBLES (MC actions, PITN-GA-2011-289-442) and the Lyon Institute of\nOrigins. The work of R.G.F. and J.P.S. was partially supported by FCT -\n\\textit{Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e a Tecnologia}, under the Projects\nPEst-OE\/FIS\/UI0777\/2011 and CERN\/FP\/123580\/2011, and by the EU RTN Marie\nCurie Project PITN-GA-2009-237920. The work of H.S. is funded by the European\nFEDER and Spanish MINECO, under the Grant FPA2011-23596.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{S:1}\n\nThis paper is concerned with scattering of time-harmonic electromagnetic plane waves by periodic structures, which is known as gratings in optics.\nThe goal is to investigate stability properties of the Helmholtz equation in both deterministic and random periodic structures.\n\nFor the scattering by a periodic structure, considerable progress has been made mathematically in literatures.\nBao, Dobson and Cox \\cite{Bao1995} reduced the scattering problem into a bounded domain problem by introducing a transparent boundary condition and proved that there exists a unique solution at all but a sequence of countable frequencies.\nLord and Mulholland \\cite{Lord2013} used the variational formula to derive a priori estimate for periodic structures, which can be viewed as an extension of \\cite{Bao1996}.\nMore results on the Helmholtz and Maxwell equations in periodic structures can be found in \\cite{Petit1980, Bao2001, Bao2021book}.\n\nRecently, there is an increasing interest in the study of wavenumber-explicit bounds for scattering problems.\nChandler-Wilde and Monk \\cite{Chandler2005} obtained a priori bounds explicit with the wavenumber for the scattering problem by unbounded rough surfaces.\nHetmaniuk \\cite{Hetmaniuk2007} established stability estimates for the Helmholtz equation with mixed boundary conditions. Esterhazy and Melenk \\cite{Esterhazy2012} further established an estimate for bounded Lipschitz domains with a Robin boundary condition.\nThe stability for the scattering problem by a large rectangular cavity was obtained by Bao, Yun and Zhou \\cite{Bao2012} in transverse electric polarization and was improved and extended to transverse magnetic polarization in \\cite{Bao2016}.\nDu, Li and Sun \\cite{LiBuyang2015} presented a numerical study of the stability estimate of the scattering by a rectangular cavity.\nWavenumber-explicit stability on the scattering problem by an obstacle in homogeneous media case \\cite{Spence2014} and heterogeneous media case \\cite{Pembery2019} were obtained\nunder the assumption that the scatterer $D$ is a star-shaped domain satisfying nontrapping conditions.\nHowever, for periodic structures, little is known about stability analysis with explicit dependence on the wavenumber, which will be derived in this paper.\nAs noted in \\cite{Bao2012, Bao2016, Chandler2005}, both the geometry and the type of boundary condition strongly affect the wavenumber-dependent stability.\nThe transparent boundary conditions (TBC) of periodic structures are different from those of open cavities and unbounded rough surfaces, which leads to additional difficulties on resonances.\nTo overcome the difficulties, a uniform distance $\\varepsilon$ is introduced in this paper and a $k$-explicit stability for periodic structures is established by a combination of the variational method and Fourier analysis.\n\n\nWe also analyze the stability for scattering by random periodic structures in this paper.\nProgress has been made recently in the exterior scattering problem by an obstacle with randomness\\cite{Spence2014, Hiptmair2018} and scattering problems in random media \\cite{Pembery2019, Pembery2020}.\nHowever, no stability result is available for scattering by random periodic structures, which has many applications in diffractive optics \\cite{Rico-Garcia2009}.\nNumerical methods for the scattering by random periodic structures have been developed recently in \\cite{Feng2018, BaoLinSINUM2020}.\nIn this work, we focus on the stability for gratings with uncertainty.\nOne difficulty is the lack of compactness. In fact, since both the deterministic and the stochastic Helmholtz equations are not coercive and the necessary compactness results are not valid any more in Bochner spaces, Fredholm theory cannot be used to the stochastic Helmholtz equation to compensate for the lack of coercivity as for the deterministic Helmholtz equation. Consequently, it is difficult to obtain the well-posedness for the random case directly as that for the deterministics case.\nIn this work, we employ Pettis measurability theorem and Bochner's theorem as in \\cite{Pembery2020} to obtain the stability estimate explicit on the wavenumber for the random case based on the stability result for the deterministic case.\nHowever, the randomness of the integral domain for the scattering by random periodic structures prevents a direct application of the general framework in \\cite{Pembery2020}, which leads to the other difficulty.\nTo overcome this difficulty, a variable transform is introduced here so that the transformed random variational form is defined on a deterministic domain with random coefficients. Similar transformation idea has also been used for other types of random differential equations \\cite{Xiu2006}.\nTherefore, for the stochastic case, by integrating the deterministic result over the probability space and introducing a transform to change the stochastic integral area into a definite one, regularity and stochastic regularity of the scattering surface, Pettis measurability theorem and Bochner's theorem yield the well-posedness and a stability estimate of our model problem.\n\nThe rest of the paper is as follows. In Section \\ref{sec:results}, the model problem is introduced, the variational formulation is described, and the results (i.e., Theorems \\ref{thm:1}-\\ref{thm:3}) are presented.\nTheorem \\ref{thm:1} concerns the stability for the Helmholtz equation in a deterministic periodic structure.\nTheorem \\ref{thm:3} gives the stability estimate for large wavenumber for the Helmholtz equation in a random periodic structure.\nThe following sections are devoted to the proofs of the two results.\nSection \\ref{sec:thm1} and Section \\ref{sec:thm23} give the detailed proofs of Theorem \\ref{thm:1} and Theorem \\ref{thm:3} respectively, followed by\nconclusion given in Section \\ref{sec:conslusion}.\n\n\n\n\n\\section{Main results}\n\\label{sec:results}\n\n\n\\subsection{Model problem}\nConsider a plane wave incident on a random periodic structure\n$$S := \\{x\\in \\mathbb{R}^2: x_2=f(\\omega;x_1),\\omega \\in \\Omega, x_1 \\in [0,\\Lambda]\\},$$\nwhich is characterized by the wavenumber $k$ ruled on a perfect conductor.\nHere, $\\omega \\in \\Omega$ denotes the random sample in a complete probability space $(\\Omega, \\mathcal{F}, \\mu)$,\n$x=(x_1,x_2)\\in \\mathbb{R}^2$ are the spatial variables, and the random surface $f: \\Omega \\times \\mathcal{X} \\rightarrow \\mathbb{R}$ is a stationary Gaussian process, each of which is a Lipschitz function, i.e. $f\\in L^2(\\Omega;Lip)$, where $Lip$ is the space of all Lipschitz functions.\nThe medium and material are assumed to be invariant in the $x_3$ direction and $\\Lambda$-periodic in the $x_1$ direction.\nThere are two fundamental polarizations for the electromagnetic fields: transverse-electric (TE) and transverse-magnetic (TM) polarization. Here, we consider the random periodic perfectly conducting grating problem for TE polarization.\n\n\\begin{figure}[h]\n\t\\centering\n\n\t\\includegraphics[width=0.9\\textwidth]{geometry.eps}\\\\\n\t\\caption{Problem geometry}\\label{fig:geometry}\n\\end{figure}\n\n\nAs shown in Figure \\ref{fig:geometry}, the grating is illuminated from above by a time-harmonic plane wave\n$\tu^{i}=e^{i\\alpha x_1-i\\beta x_2},$\nwhere\n$ \\alpha = k \\sin\\theta,~\\beta=k \\cos\\theta$,\n$\\theta\\in(-\\pi\/2,\\pi\/2)$ is the incident angle with respect to the positive $x_2$-axis.\nDenote $$D^+= \\{x\\in\\mathbb{R}^2: x_2>f(\\omega; x_1),\\ \\omega \\in \\Omega, x_1 \\in [0,\\Lambda]\\}.$$ Since the medium below $S$ is perfectly electric conducting, the scattering problem for TE polarization can be modeled by the following two-dimensional Helmholtz equation with the homogeneous boundary condition:\n\\begin{eqnarray}\\label{eq:u0}\n\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta u(\\omega ; \\cdot)+k^{2} u(\\omega ; \\cdot)=0 & \\text { in } \\Omega \\times D^+, \\\\\n\t\tu(\\omega;\\cdot)=0 & \\text { on } \\Omega \\times S ,\n\t\\end{array}\\right.\n\\end{eqnarray}\nwhere $u(\\omega ; \\cdot)$ is the total field.\n\n\\subsection{Variational form}\n\\label{sec:variational}\nIn order to obtain a stability estimate, an important step is to reduce the infinite scattering problem into a bounded domain problem by introducing a transparent boundary condition \\cite{Bao1995SINUM}.\n\nSince the total field can be decomposed into\n$\nu(\\omega;\\cdot) = u^{i} + u^s(\\omega;\\cdot)\n$\nwhere\nthe incident wave satisfies\n\\begin{equation*}\n\t\\Delta u^{i}+k^{2} u^{i}=0 \\quad \\text { in } \\Omega \\times D^+,\n\\end{equation*}\nthe scattered field $u^s(\\omega;\\cdot)$ satisfies\n\\begin{eqnarray}\\label{eq:u}\n\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta u^s(\\omega ; \\cdot)+k^{2} u^s(\\omega ; \\cdot)=0 & \\text { in } \\Omega \\times D^+, \\\\\n\t\tu^s(\\omega ; \\cdot)+u^{i}=0 & \\text { on } \\Omega \\times S .\n\t\\end{array}\\right.\n\\end{eqnarray}\n\nDenote by $\\Gamma = \\{ (x_1,x_2)\\in\\mathbb{R} ^2 : x_2=b, x_1 \\in [0,\\Lambda] \\} $\nwith\n$b > \\max_{\\omega\\in \\Omega, x_1\\in(0,\\Lambda)} f(\\omega;x_1). $\nThe scattered field $u^s$ above $\\Gamma$ admits the Rayleigh expansion\n\\begin{equation}\\label{Rayleigh}\n\tu^s(\\omega;x_1,x_2)=\\sum_{n \\in Z} A_{n} e^{i\\left(\\alpha_{n} x_1+\\beta_{n} x_2\\right)} ,\n\\end{equation}\nwhere\n\\begin{equation}\\label{alpha_n}\n\n\t\\alpha_{n}=\\alpha+\\dfrac{2 \\pi n}{\\Lambda}, \\beta_{n}^{2}=k^{2}-\\alpha_{n}^{2} \\text { with } \\Im \\beta_{n} \\geq 0.\n\\end{equation}\n\n\\begin{assumption}\\label{assump:k}\n\t(Exclusion of resonances). Assume there exists $\\varepsilon>0$, such that $\\vert k-\\alpha_n \\vert\\geq \\varepsilon, \\ \\forall n \\in \\mathbb{Z} $.\n\\end{assumption}\n\n\\begin{remark}\n\tA constant $\\varepsilon$ is introduced here to exclude\tall possible resonances ($\\kappa = \\alpha_n$) uniformly and thus a stability explicit with the wavenumber $k$ is obtained by a combination of the variational method and Fourier analysis.\n\\end{remark}\n\n\n\nDefine a boundary operator $T:H^{\\frac{1}{2}}(\\Gamma) \\rightarrow H^{-\\frac{1}{2}}(\\Gamma)$ by\n\\begin{equation*}\n\t(T v)\\left(x_{1} \\right)=\\sum_{n \\in \\mathbb{Z} } \\mathrm{i} \\beta_{n} v_{n} e^{\\mathrm{i} \\alpha_{n} x_{1} } ,\n\\end{equation*}\nwhere\n$v_{n} (\\omega)=\\frac{1} {\\Lambda} \\int_{0} ^{\\Lambda} v\\left(\\omega;x_{1} ,b\\right) e^{-\\mathrm{i} \\alpha_{n} x_{1} } \\mathrm{d} x_{1} , $\nwhich yields a transparent boundary condition on $\\Gamma$:\n\\begin{equation*}\n\t\\partial_{\\nu} u = T u +g(x_1) \\quad \\text{ on } \\Gamma,\n\\end{equation*}\nand $g(x_1)= \\partial_{\\nu} u^{i} - T u^{i} = -2 i \\beta e^{i \\alpha x_1 - i \\beta b}$.\n\nDenote by $D= \\{x\\in\\mathbb{R}^2: f(\\omega; x_1)1$ and large $k$, such that\n\t\\begin{equation}\\label{Destimate}\n\t\t\\|\\nabla \\tilde{u} \\|_{L^2(D)} + k \\|\\tilde{u}\\|_{L^2(D)} \\leq C \\max \\left\\{ \\dfrac{ b^2 {k} ^{2} } {\\sqrt{\\varepsilon} } ,b^3 {k} ^{\\frac{5} {2} } \\right\\} \\|g\\|_{L^2(\\Gamma)}.\n\t\\end{equation}\n\\end{theorem}\n\n\n\\begin{remark} \\label{re:1}\n\tFor the estimate, $\\varepsilon$ is independent of the wavenumber $k$ and the measured height $b$. Note that for small \t$\\varepsilon$, wavenumber is close to resonances and then the corresponding bound is very large.\n\n\tIf $\\varepsilon\\geq \\frac{1} { b^2 k } $, then the estimate becomes\n\t\\begin{equation}\\label{reduced}\n\t\t\\|\\nabla \\tilde{u} \\|_{L^2(D)} + k \\| \\tilde{u} \\|_{L^2(D)} \\leq C b^3 k ^{\\frac{5} {2} } \\|g\\|_{L^2(\\Gamma)}.\n\t\\end{equation}\n\\end{remark}\n\nNext, the first stability result in a random periodic structure is shown in Theorem \\ref{thm:3} under Assumption \\ref{assump:k} and Assumption \\ref{cond:f}.\n\n\\begin{assumption}\\label{cond:f}\n\t(Regularity and stochastic regularity of $f$). Assume the random structure $ f\\in L^2(\\Omega;Lip)$.\n\\end{assumption}\n\n\\begin{remark}\\label{remark:f0}\n\t(Random surfaces satisfying Assumption \\ref{cond:f}). Motivated by the uncertainty quantification such as Karhunen-Lo\\`eve expansion and other similar expansions, it makes sense to consider $f$ as series expansions around the known deterministic surface $f_0$ satisfying the Lipschitz condition.\n\\end{remark}\n\n\\begin{theorem}\\label{thm:3}\n\t(Stability for the random case).\n\tUnder Assumption \\ref{assump:k} and Assumption \\ref{cond:f},\n\tthere exists a unique solution $u\\in L^2(\\Omega; H_{S,qp}^1(D))$ to Problem \\ref{P3}.\n\tMoreover, there exists a constant $C$ independent of $b>1$ and large $k$, such that\n\t\\begin{equation}\\label{Sestimate}\n\t\t\\|\\nabla u \\|_{L^2(\\Omega;L^2(D))} + k \\|u\\|_{L^2(\\Omega;L^2(D))} \\leq C \\max \\left\\{ \\dfrac{b^2 {k} ^{2} } {\\sqrt{\\varepsilon} }\n\t\t,b^3 {k} ^{\\frac{5} {2} } \\right\\} \\|g\\|_{L^2(\\Omega;L^2(\\Gamma))}.\n\t\\end{equation}\n\\end{theorem}\n\n\n\\section{Stability for the deterministic case}\n\\label{sec:thm1}\n\nThis section is devoted to the proof of Theorem \\ref{thm:1}. Our approach based on the variational method and Fourier analysis consists of two steps: the first step is to estimate the norms of $u$ in the domain $D$ by the norms of $u$ on the boundary $\\Gamma$ and the second step is to estimate the norms of $u$ on $\\Gamma$ by the norm of $g$.\n\nAlthough our approach is related to \\cite{Bao2012,Bao2016} for the scattering by a large rectangular cavity, additional difficulties arise: (1) Because of the different model geometry, quasiperiodic solutions have to be considered for our model here. (2) Instead of the Dirichlet or Neumann boundary condition for the cavity problem, the (nonlocal) transparent boundary conditions must be dealt with. (3) Unlike the cavity problem, our model problem may not have unique solutions due to the resonances. Therefore, new techniques must be developed to prove the stability result.\n\nBefore proceeding to the detailed proof, some useful notations and preliminary results are introduced first.\n\n\n\n\\subsection{Notations and preliminary results}\n\\label{S:preliminary}\n\nLet $\\tilde{u}(x_1,x_2)=u(\\omega_0;x_1,x_2)$ be the solution to the deterministic problem for a given sample $\\omega_0$. Then the total field $\\tilde{u}$ satisfies\n\\begin{equation}\\label{eq:u_deter}\n\t\\left\\{\\begin{array}{ll}\n\t\t\\Delta \\tilde{u}+k^{2} \\tilde{u}=0 & \\text { in } D, \\\\\n\t\t\\tilde{u}=0 & \\text { on } S, \\\\\n\t\t\\partial_{\\nu}\\tilde{u} = T\\tilde{u}+g & \\text{ on } \\Gamma.\n\t\\end{array}\\right.\n\\end{equation}\n\nLet $C$ denote positive constants which are independent of large $k$ and $b>1$. Denote by $ \\alpha \\simeq \\beta$ if there exists a positive constant $C$ independent of large $k$ and $b>1$ such that $\\frac{1}{C} \\alpha \\leq \\beta\\leq C \\alpha$.\nWithout loss of generality, assume the period $\\Lambda = 2\\pi$ and denote $\\tilde{u}$ as $u$ for simplicity.\n\n\nThe scattered field $u$ can be written in the single-layer potential representation:\n\\begin{equation}\\label{u1}\n\tu(x_1,x_2)=\\int_{0}^{\\Lambda} \\phi(s) G(x_1, x_2 ; s, 0) \\mathrm{d} s,\n\\end{equation}\nwhere\n$\\phi \\in L^2(0,\\Lambda)$ is an unknown periodic density function and $G$ is a quasi-periodic Green function given explicitly as\n$$\nG(x_1, x_2 ; s, t)=\\frac{i}{2 \\pi} \\sum_{n \\in \\mathbb{Z}} \\frac{1}{\\beta_{n}} e^{i \\alpha_{n}(x_1-s)+i \\beta_{n}\\vert x_2-t\\vert }, \\quad(x_1,x_2) \\neq(s, t).\n$$\nWithout loss of generality, it is assumed that the normal incidence is with $2 \\pi$-periodicity, i.e.\n$\\theta=0, \\alpha=0, \\beta=k, \\Lambda = 2 \\pi.$\nTherefore\n$\\alpha_n=n$ and $\\beta_n=\\sqrt{k^2-n^2}$,\nand thus\n\\begin{equation}\\label{Gn}\n\tG(x_1,x_2; s, t)=\\frac{i}{2 \\pi} \\sum_{n \\in \\mathbb{Z}} \\frac{1}{\\sqrt{k^2-n^2}} e^{i n(x-s)+i \\sqrt{k^2-n^2}\\vert y-t\\vert }\n\n\t\\text{~~and~~}\n\n\t\\phi(s)=\\sum_{n \\in \\mathbb{Z}} \\phi_{n} e^{i n s}.\n\\end{equation}\nSubstituting \\eqref{Gn} into \\eqref{u1} yields\n$$\nu^s (x_1,x_2)=\\sum_{n \\in \\mathbb{Z}} \\psi_n e^{i nx_1+i \\sqrt{k^2-n^2} x_2}.\n$$\nSince $u=u^{inc} +u^s$, the solution $u$ can be expressed as\n\\begin{equation*}\n\t\\begin{aligned}\n\t\tu(x_1,x_2)=&\n\t\t\\displaystyle\\sum_{n=0} ^{\\infty} a_{n} \\sin \\left(\\sqrt{{k} ^{2} -n^{2} } x_2\\right) \\sin (n x_1) + \\sum_{n=0} ^{\\infty} b_{n} \\cos \\left(\\sqrt{{k} ^{2} -n^{2} } x_2\\right) \\cos (n x_1)\\\\\n\t\t& + i \\left[ \\displaystyle\\sum_{n=0} ^{\\infty} c_n \\cos \\left(\\sqrt{{k} ^{2} -n^{2} } x_2\\right) \\sin (n x_1) + \\sum_{n=0} ^{\\infty} d_{n} \\sin \\left(\\sqrt{{k} ^{2} -n^{2} } x_2\\right) \\cos (n x_1)\\right]\\\\\n\t\t\\triangleq& \\mu+i \\nu.\n\t\\end{aligned}\n\\end{equation*}\nwhere $a_n=-\\psi_n+\\psi_{-n} $, $b_n=\\psi_n+\\psi_{-n} $, $n\\in \\mathbb{N} _+$, $a_0=-\\psi_0+1$, $b_0=\\psi_0+1$; $c_n=-a_n$, $d_n=b_n$, $n\\in \\mathbb{N} _+$, $c_0=\\psi_0-1$, $d_0=\\psi_0+1$.\nFor the case of non-normal cases, we only need to replace $n$ with $\\alpha_n$ and $\\sqrt{k^2-n^2} $ with $\\sqrt{k^2-{\\alpha_n} ^2} $.\nSince $\\mu$ and $\\nu$ can be studied similarly, only detailed study on $\\mu$ is provided here. Also, for convenience, in the following proof, the notation $\\mu$ is denoted as $u$.\n\nLet $u_n$, $(\\partial_{\\nu}u)_n$, $(Tu)_n$ be the sine coefficients, and $v_n$, $(\\partial_{\\nu}v)_n$, $(Tv)_n$ be the cosine coefficients of the expansions of $u$, $\\partial_{\\nu}u$, $Tu$, on $\\Gamma$, respectively. That is,\n\\begin{equation}\\label{unvn}\n\tu_{n}:=\n\ta_{n} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right),\n\tv_{n}:=\n\tb_{n} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right),\n\\end{equation}\n\\begin{equation}\\label{partialun}\n\t\\left(\\partial_{\\nu} u\\right)_{n}:=\n\ta_{n} \\sqrt{k^{2}-n^{2}} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right),\n\\end{equation}\n\\begin{equation}\\label{partialvn}\n\t\\left(\\partial_{\\nu} v\\right)_{n}:=\n\t-b_{n} \\sqrt{k^{2}-n^{2}} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right).\n\\end{equation}\nHence\n\\begin{equation*}\n\tu\\vert_{\\Gamma}(x_1) = \\sum_{n=0}^{\\infty} u_n \\sin(n x_1) + \\sum_{n=0}^{\\infty} v_n \\cos(n x_1),\n\\end{equation*}\n\\begin{equation*}\n\t\\partial_{\\nu} u\\vert_{\\Gamma}(x_1) = \\sum_{n=0}^{\\infty} (\\partial_{\\nu} u)_n \\sin(n x_1) + \\sum_{n=0}^{\\infty} (\\partial_{\\nu} v)_n \\cos(n x_1),\n\\end{equation*}\nand\n\\begin{equation*}\n\tTu\\vert_{\\Gamma}(x_1) = \\sum_{n=0}^{\\infty} (Tu)_n \\sin(n x_1) + \\sum_{n=0}^{\\infty} (Tv)_n \\cos(n x_1).\n\\end{equation*}\n\n\n\\begin{definition}\\label{def:AB}\n\tFor $u =\\displaystyle\\sum_{n=0}^{\\infty} u_n \\sin(n x_1) + \\sum_{n=0}^{\\infty} v_n \\cos(n x_1)$, define\n\t$$\\|u\\|_{A}=\\sqrt{\\sum_{n=0}^{\\infty}\\left( \\lvert u_{n}\\rvert^{2} +\\lvert v_{n}\\rvert^{2} \\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}}},$$\n\tand\n\t$$\\|u\\|_{B}=\\sqrt{\\sum_{n=0}^{\\infty}\\left( \\lvert u_{n}\\rvert^{2} +\\lvert v_{n}\\rvert^{2}\\right) \\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+ \\frac{1}{b}\\right)}.$$\nHere, the norms $A$ and $B$ are formerly dual to each other and behave like $H^{-1\/2}$, $H^{1\/2}$, respectively. Moreover, a simple calculation gives that\n\t\\begin{equation}\\label{norm}\n\t\t\\int_{\\Gamma} u_1(x_1) \\overline{u_2(x_1)} \\mathrm{d} x_1 \\leq \\pi \\|u_1\\|_{A}\\|u_2\\|_{B}.\n\t\\end{equation}\n\\end{definition}\n\n\n\\begin{definition}\n\tDefine the lower frequency set $\\mathbf{L}$ and the higher frequency set $\\mathbf{H}$ by\n\t$$\\mathbf{L}=\\left\\{n \\in \\mathbb{N} \\mid n < k \\right\\}\n\t\\text{~and~~}\n\t\\mathbf{H}=\\left\\{n \\in \\mathbb{N} \\mid n > k \\right\\}.$$\n\\end{definition}\n\nThen, $u$, $\\partial_{\\nu}u$, $T(u)$ on $\\Gamma$ defined above can be separated into the lower frequency part and the higher frequency part:\n$$\nu(x_1) =\\mathcal{V}_\\mathbf{L} +\\mathcal{V}_\\mathbf{H}, \\quad\n\\partial_{\\nu} u(x_1) =\\mathcal{P}_\\mathbf{L} +\\mathcal{P}_\\mathbf{H},\\quad\n(T (u))(x_1) = \\mathcal{T}_\\mathbf{L} +\\mathcal{T}_\\mathbf{H},$$\nwith\n\\begin{equation*}\n\t\\begin{aligned}\n\t\t&\\mathcal{V}_\\mathbf{L} =\\sum_{n \\in \\mathbf{L}} u_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{L}} v_{n} \\cos (n x_1),\\\\\n\t\t&\\mathcal{V}_\\mathbf{H}=\\sum_{n \\in \\mathbf{H}} u_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{H}} v_{n} \\cos (n x_1), \\\\\n\t\t&\\mathcal{P}_\\mathbf{L}=\\sum_{n \\in \\mathbf{L}}\\left(\\partial_{\\nu} u\\right)_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{L}}\\left(\\partial_{\\nu} v\\right)_{n} \\cos (n x_1),\\\\\n\t\t&\\mathcal{P}_\\mathbf{H}=\\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n} \\cos (n x_1),\\\\\n\t\t&\\mathcal{T}_\\mathbf{L} =\\sum_{n \\in \\mathbf{L}}(T u)_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{L}}(T v)_{n} \\cos (n x_1),\\\\\n\t\t&\\mathcal{T}_\\mathbf{H}=\\sum_{n \\in \\mathbf{H}}(T u)_{n} \\sin (n x_1)\n\t\t+\\sum_{n \\in \\mathbf{H}}(T v)_{n} \\cos (n x_1).\n\t\\end{aligned}\n\\end{equation*}\n\n\nThe higher frequency terms have useful properties presented in Lemma\n\\ref{Lem:H}.\n\\begin{lemma}\\label{Lem:H}\n\tIf $u(x_1,x_2)$ is the solution to the scattering problem \\eqref{eq:u_deter}, then\n\t$$ \\left\\|\\mathcal{P}_\\mathbf{H}\\right\\|_{A}^{2}\n\t\\simeq \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}} \\simeq\\left\\|\\mathcal{V}_\\mathbf{H}\\right\\|_{B}^{2} .$$\n\\end{lemma}\n\\begin{proof}\n\tIf $n\\in \\mathbf{H}$,\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t&\\left( \\lvert a_{n} \\sqrt{k^{2}-n^{2}} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\rvert^2 + \\lvert -b_{n} \\sqrt{k^{2}-n^{2}} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\rvert^2 \\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}}\\\\\n\t\t\t\\simeq & \\left( \\lvert a_{n} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\rvert^2 + \\lvert b_{n} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\rvert^2 \\right) \\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+ \\frac{1}{b}\\right).\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tHence\n\t\\begin{equation*}\n\t\t\\left( \\lvert (\\partial_{\\nu}u)_n \\rvert^2 + \\lvert (\\partial_{\\nu}v)_n \\rvert^2 \\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}}\n\t\t\\simeq \\left( \\lvert u_n \\rvert^2 + \\lvert v_n \\rvert^2 \\right) \\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+ \\frac{1}{b}\\right).\n\t\\end{equation*}\n\tFrom \\eqref{unvn}-\\eqref{partialvn}, one can show that for any $n \\in \\mathbf{H}$,\n\t$\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} \\geq 0$, $\\left(\\partial_{\\nu} v\\right)_{n} \\overline{v_{n}} \\geq 0$.\n\tTherefore,\n\t$$ \\left\\|\\mathcal{P}_\\mathbf{H} \\right\\|_{A}^{2}\n\t\\simeq \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}} .$$\n\\end{proof}\n\nThe next result provides a quantitative estimate of the function $u$ and its normal derivative.\n\n\\begin{lemma}\\label{Lem:sqrtk}\n\tIf $u(x_1,x_2)$ is the solution to the scattering problem \\eqref{eq:u_deter}, then\n\t\\begin{equation}\\label{Lemineq}\n\t\t\\Lambda \\sum_{n\\in \\mathbb{Z} } \\sqrt{\\vert {k} ^{2} -\\alpha_n^2\\vert} \\vert\\hat{u} _n\\vert^2 \\geq \\sqrt{\\varepsilon} \\sqrt{k} \\|u\\|_{L^2(\\Gamma)} ^2.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\tSince $\\Lambda \\sum_{n\\in \\mathbb{Z} } \\vert\\hat{u} _n\\vert^2 = \\|u\\|_{L^2(\\Gamma)} ^2$ and\n\t$\\sqrt{\\vert {k} ^{2} -\\alpha_n^2 \\vert} \\geq \\sqrt{\\varepsilon} \\sqrt{k} $ for $\\vert k-\\alpha_n\\vert\\geq \\varepsilon, \\ \\forall n \\in \\mathbb{Z} $,\n\t$$\\Lambda \\sum_{n\\in \\mathbb{Z} } \\sqrt{\\vert{k} ^{2} -\\alpha_n^2\\vert} \\vert\\hat{u} _n\\vert^2 \\geq \\sqrt{\\varepsilon} \\sqrt{k} \\|u\\|_{L^2(\\Gamma)} ^2,$$\n\twhich completes the proof.\n\\end{proof}\n\n\\begin{remark}\n\tDifferent from Lemma 3.5 in \\cite{Bao2012} for the scattering by a cavity, TBC of gratings yield additional difficulties on resonances and stability.\n\tHence, in this paper, a uniform distance $\\varepsilon$ is introduced to exclude resonances and thus obtain the estimate \\eqref{Lemineq} in Lemma \\ref{Lem:sqrtk}, which is essential for the proof procedure of the $k$-explicit stability for periodic structures.\n\\end{remark}\n\nNext, the relation between the norm of $A$ (or $B$) and the $L^2$ norm is established in Remark \\ref{rem:AL}.\n\n\\begin{remark}\\label{rem:AL}\n\tFor the norm $A$, if there exists a natural number $n^*$ such that $\\sqrt{\\vert k^2-{n^*}^2 \\vert}\\leq \\frac{1}{b}$, then $n^*$ is very close to $k$ so that $\\vert k-{n^*} \\vert\\leq \\frac{1}{ b^2 k}$. Since $\\sqrt{\\vert k^2-n^2 \\vert}\\geq \\sqrt{\\frac{k}{2}}$,\n\tthe norm $A$ can be simplified as\n\t$$\\|u\\|_{A} \\simeq \\sqrt{\\left(\\sum_{n \\in \\mathbb{N} \\backslash\\left\\{n^{*}\\right\\}}\\left(\\lvert u_{n}\\rvert^{2}+\\lvert v_{n}\\rvert^{2}\\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}}\\right)+\\left(\\lvert u_{n^*}\\rvert^{2}+\\lvert v_{n^*}\\rvert^{2}\\right) b}.$$\t\n\tIf such $n^*$ does not exist, then the norm $A$ has no relation to $b$ since\n\t$$\\|u\\|_A \\simeq \\sqrt{\\sum_{n \\in \\mathbb{N}}\\left(\\lvert u_{n}\\rvert^{2}+\\lvert u_{n}\\rvert^{2}\\right) \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}}}.$$\n\tHence, whether $n^*$ exists or not, there always exists a constant $C$ independent of $k$ and $b$ such that\n\t$$\\|u\\|_{A}^{2} \\leq C\\left(\\frac{1}{\\sqrt{k}}\\|u\\|_{L^{2}(\\Gamma)}^{2}+b \\left( \\lvert u_{k^{*}}\\rvert^{2}+\\lvert v_{k^{*}}\\rvert^{2}\\right)\\right),$$\n\twhere $k^*=\\arg\\min_{n\\in\\mathbb{N}} \\vert n-k\\vert $, $u_{k^*} = \\frac{1}{\\pi} \\int_{\\Gamma} u(x_1) \\sin \\left(k^{*} x_1\\right) \\mathrm{d} x_1 $ and $v_{k^*} = \\frac{1}{\\pi} \\int_{\\Gamma} u(x_1) \\cos \\left(k^{*} x_1\\right) \\mathrm{d} x_1$.\n\tIt follows immediately that\n\t\\begin{equation}\n\t\t\\|u\\|_{A}^{2} \\leq C b\\|u\\|_{L^{2}(\\Gamma)}^{2}.\n\t\\end{equation}\n\tThe similar approximation can be deduced for the norm $B$:\n\t\\begin{equation}\\label{ineq:normB}\n\t\tb\\|u\\|_{B}^{2} \\leq C \\|u\\|_{L^{2}(\\Gamma)}^{2}.\n\t\\end{equation}\n\\end{remark}\n\nAs for the proof of Theorem \\ref{thm:1},\nthe first step given in Lemma \\ref{lem:step1} is to estimate the norm of $u$ in $D$ by the norms of $u$ on $\\Gamma$, i.e., the norms $A$ and $B$, and the second step given in Lemma \\ref{lem:step2} is to estimate $\\|\\partial_{\\nu}u\\|_A$ and $\\|u\\|_B$ by $\\|g\\|_A$. Finally, with the correlation between the norm $A$ and the $L^2$ norm discussed in Remark \\ref{rem:AL}, the stability estimate of Theorem \\ref{thm:1} is proved.\n\n\\begin{lemma}\\label{lem:step1}\n\tLet $u$ be a solution to the scattering problem \\eqref{eq:u_deter}. Under Assumption \\ref{assump:k}, there exists a constant $C$ independent of $b>1$ and large $k$, such that\n\t\\begin{equation}\n\t\t\\|\\nabla u \\|_{L^2(D)} + k \\|u\\|_{L^2(D)} \\leq C b k \\left( \\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\right),\n\t\\end{equation}\n\twhere the norms $A$ and $B$ are defined in Definition \\ref{def:AB}.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:step2}\n\tLet $u$ be a solution to the scattering problem \\eqref{eq:u_deter}. Under Assumption \\ref{assump:k}, there exists a constant $C$ independent of $b>1$ and large $k$, such that\n\t\\begin{equation}\n\t\t\\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\leq C \\max \\left\\{ \\dfrac{ b^{\\frac{1} {2} } {k} } {\\sqrt{\\varepsilon} }\n\t\t, b^{\\frac{3} {2} } {k} ^{\\frac{3} {2} } \\right\\} \\|g\\|_A,\n\t\\end{equation}\n\twhere the norms $A$ and $B$ are defined in Definition \\ref{def:AB}.\n\\end{lemma}\n\n\\subsection{Proof of Lemma \\ref{lem:step1} and Lemma \\ref{lem:step2}}\n\n\nThe proof of the bound in Lemma \\ref{lem:step1} is mainly based on the definition of the norms and the representation of the series, which means that a stability estimate for the solution $u$ in $D$ can be given in terms of its Dirichlet and Neumann data on $\\Gamma$.\n\n\\begin{proof}[Proof of Lemma~{\\upshape\\ref{lem:step1}}]\n\tFrom the equation $\\Delta u + k^2 u = 0 $ in $D$ and its boundary condition on $S$\n\n\tas well as the quasi-periodicity of the solution $u$, one has\n\t$$ \\int_{\\Gamma} u \\overline{\\partial_{\\nu} u} \\mathrm{d} x_1=\\int_{D}\\left(\\vert\\nabla u\\vert^{2}-k^{2}\\vert u\\vert^{2}\\right) \\mathrm{d} x_1 \\mathrm{d} x_2. $$\n\t\n\tThe left-hand side of the equality above is obviously bounded by $ \\left\\|\\partial_{\\nu} u\\right\\|_{A}^{2}+\\|u\\|_{B}^{2}$.\n\tHowever, the right-hand side has a negative sign for the term $\\vert u\\vert^2$. In order to prove\n\tLemma \\ref{lem:step1}, it suffices to show that\n\t\\begin{equation}\\label{6.1}\n\t\t\\|u\\|_{L^{2}(D)}^{2} \\leq C b^{2}\\left(\\left\\|\\partial_{\\nu} u\\right\\|_{A}^{2}+\\|u\\|_{B}^{2}\\right).\n\t\\end{equation}\n\t\n\tNote that $\\partial_{\\nu}u$, $u$ on $\\Gamma$, and $u$ in $D$ can be expressed in terms of sine and cosine functions in the direction of $x_1$. Because of the orthogonality of sine and cosine functions, one may assume\n\tthat\n\t$$ u(x_1,x_2)=\\eta \\sin \\left(\\sqrt{k^{2}-n^{2}} x_2\\right) \\sin (n x_1) + \\zeta \\cos \\left(\\sqrt{k^{2}-n^{2}} x_2\\right) \\cos (n x_1)\\text { in } D, \\ n\\neq k. $$\n\n\tFor $n \\neq k$, consider three cases:\n\t(i) $nk$ and $b \\sqrt{\\vert k^2-n^2 \\vert}\\geq 1$.\n\t\n\tFor Case (i), it is easy to see that $\\|u\\|_{L^{2}(D)}^{2} \\simeq b$. On the boundary $\\Gamma$,\n\t$$\n\t\\begin{aligned}\n\t\t\\partial_{\\nu} u &=\\eta \\sqrt{k^{2}-n^{2}} \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\sin (n x_1) - \\zeta \\sqrt{k^{2}-n^{2}} \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\cos (n x_1), \\\\\n\t\tu &=\\eta \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\sin (n x_1) + \\zeta \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right) \\cos (n x_1).\n\t\\end{aligned}\n\t$$\n\tFrom the definition of the norms A and B, one has\n\t$$\n\t\\begin{array}{l}\n\t\t\\left\\|\\partial_{\\nu} u\\right\\|_{A}^{2}+\\|u\\|_{B}^{2} \\\\\n\t\t\\qquad \\begin{aligned}\n\t\t\t=& \\vert\\eta\\vert^2 \\left(\\frac{k^{2}-n^{2}}{\\sqrt{k^{2}-n^{2}}+\\frac{1}{b}}\\lvert \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right)\\rvert^{2}\n\t\t\t+\\left(\\sqrt{k^{2}-n^{2}}+\\frac{1}{b}\\right)\\lvert \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right)\\rvert^{2} \\right)\\\\\n\t\t\t& + \\vert\\zeta\\vert^2 \\left(\\frac{k^{2}-n^{2}}{\\sqrt{k^{2}-n^{2}}+\\frac{1}{b}}\\lvert \\sin \\left(\\sqrt{k^{2}-n^{2}} b\\right)\\rvert^{2}\n\t\t\t+\\left(\\sqrt{k^{2}-n^{2}}+\\frac{1}{b}\\right)\\lvert \\cos \\left(\\sqrt{k^{2}-n^{2}} b\\right)\\rvert^{2} \\right)\\\\\n\t\t\t\\geq & \\frac{1}{2} (\\vert\\eta\\vert^2+\\vert\\zeta\\vert^2)\\sqrt{k^{2}-n^{2}} \\geq \\frac{(\\vert\\eta\\vert^2+\\vert\\zeta\\vert^2)}{2 b} \\geq C \\frac{1}{b^{2}}\\|u\\|_{L^{2}(D)}^{2}.\n\t\t\\end{aligned}\n\t\\end{array}\n\t$$\n\t\n\tFor Case (ii), one has\n\t$$ \\lvert \\sin \\left(\\sqrt{k^{2}-n^{2}} x_2\\right)\\rvert \\simeq \\sqrt{\\lvert k^{2}-n^{2}\\rvert} x_2 \\text{ and }\n\t\\lvert \\cos \\left(\\sqrt{{k} ^{2} -n^{2} } x_2\\right)\\rvert \\simeq 1- \\dfrac{1} {2} \\lvert {k} ^{2} -n^{2} \\rvert x_2^2$$\n\tfrom $ \\sqrt{\\lvert k^{2}-n^{2}\\rvert} b<1 $.\n\tTherefore, $\\|u\\|_{L^{2}(D)}^{2} \\simeq\\lvert k^{2}-n^{2}\\rvert b^{3}$ and\n\t$$ \\left\\|u\\left(x, b\\right)\\right\\|_{B}^{2} \\simeq\\left(\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}\\right)\\lvert k^{2}-n^{2}\\rvert b^{2} \\simeq b\\lvert k^{2}-n^{2}\\rvert, $$\n\twhich implies the inequality \\eqref{6.1}.\n\t\n\tFinally, for Case (iii),\n\t$$\n\t\\begin{aligned}\n\t\tu&=\\eta \\sin \\left(\\sqrt{k^{2}-n^{2}} x_2\\right) \\sin (n x_1) + \\zeta \\cos \\left(\\sqrt{k^{2}-n^{2}} x_2\\right) \\cos (n x_1) \\\\\n\t\t&= \\eta i \\sinh \\left(\\sqrt{n^{2}-k^{2}} x_2\\right) \\sin (n x_1) + \\zeta \\cosh \\left(\\sqrt{n^{2}-k^{2}} x_2\\right) \\cos (n x_1) \\text { in } D.\n\t\\end{aligned}\n\t$$\n\tThe representation of $u$ yields\n\t$$ \\|u\\|_{L^{2}(D)}^{2} \\leq C \\frac{1}{\\sqrt{n^{2}-k^{2}}} e^{2 \\sqrt{n^{2}-k^{2}} b} \\leq C b^{2}\\left(\\sqrt{n^{2}-k^{2}}+\\frac{1}{b}\\right) e^{2 \\sqrt{n^{2}-k^{2}} b} \\leq C b^{2}\\left\\|u\\left(\\cdot, b\\right)\\right\\|_{B}^{2}. $$\n\\end{proof}\n\n\nFor the proof of Lemma \\ref{lem:step2}, the main difficulty is due to the fact that the boundary data $\\partial_{\\nu} u = T u +g$ does not yield direct estimates on $\\| \\partial_{\\nu}u \\|_A$ and $\\|u\\|_B$. To be specific, let us consider the key term $-\\mathfrak{Im}\\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1$. Obviously,\n\\begin{equation}\\label{diff}\n\t- \\mathfrak{Im}\\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1 = \\Lambda \\sum_{\\vert \\alpha_n \\vert k $.\nIn order to obtain the stability, we have to analyze $\\hat{u}$ for $\\vert \\alpha_n \\vert > k $, and thus to generate a bound on $u$. Next, we give the proof of Lemma \\ref{lem:step2}.\n\n\\begin{proof}[Proof of Lemma~{\\upshape\\ref{lem:step2}}]\n\tFrom the definition,\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1 & = \\int_{\\Gamma} (\\partial_{\\nu} u - T(u)) \\bar{u} \\mathrm{d} x_1\\\\\n\t\t\t& = \\int_{\\Gamma} \\partial_{\\textbf{n} } u \\bar{u} \\mathrm{d} x_1- i \\Lambda \\sum_{\\vert n \\vertk} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2 \\\\\n\t\t\t& \\triangleq \\mathfrak{Re} + i \\mathfrak{Im},\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twith\n\t$\\mathfrak{Re} = \\int_{\\Gamma} \\partial_{\\textbf{n} } u \\bar{u} \\mathrm{d} x_1 + \\Lambda \\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2$ and\n\t$ \\mathfrak{Im} = - \\Lambda \\sum_{\\vert n \\vertk} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2,\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tone obtains\n\t\\begin{equation}\\label{case2Re}\n\t\t\\begin{aligned}\n\t\t\t&\\gamma \\pi \\left\\| \\partial_{\\nu}u \\right\\|_A \\|u\\|_B + \\lvert \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\rvert \\\\\n\t\t\t& \\geq \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}} \n\t\t\t + \\Lambda \\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2,\n\t\t\\end{aligned}\n\t\\end{equation}\n\twith\n\t$$ \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}} \n\t \\simeq \\left\\| \\mathcal{P}_\\mathbf{H} \\right\\|_A^2 \\geq 0$$\n\tand\n\t$$\\Lambda \\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2 \\geq 0.\n\t$$\n\n\tThe estimate \\eqref{case2Re} implies that $\\displaystyle\\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} u\\right)_{n} \\overline{u_{n}} + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}} $ and $ \\Lambda \\displaystyle\\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2$ can be bounded by $\\lvert \\displaystyle\\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\rvert$ \nsince $\\gamma \\pi \\left\\| \\left( \\partial_{\\nu}u \\right)_n \\right\\|_A \\|u\\|_B$ may be absorbed by choosing an appropriate $\\gamma$.\n\nEvidently, in order to estimate $ \\Lambda \\displaystyle\\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2$, it is sufficient to estimate $\n\t\\lvert \\displaystyle\\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\rvert\\;.\n\t$\n\tSince the estimate \\eqref{case2Re} is based on the condition \\eqref{condcase2}, we will consider $\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u \\|_A$ and $\\| g \\|_A \\geq \\gamma \\| \\partial_{\\nu}u \\|_A$ separately. For the case $\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u \\|_A$, since the term $\n\t\\lvert \\displaystyle\\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{L}} \\left( \\partial_{\\nu}v \\right)_n \\bar{v}_n \\rvert\n\t$ concerns the low frequency part for ${u}_n$ and $\\left( \\partial_{\\nu}u \\right)_n$, to characterize which term is bigger, $\\delta \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A \\geq \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B$ and $\\delta \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A \\leq \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B$ are considered separately. \t\n\tTo sum up, we consider the following three cases:\n\t\\begin{itemize}\n\t\t\\item Case (i)\n\t\t\\begin{equation}\\label{case2.1}\n\t\t\t\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u \\|_A, \\quad \\delta \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A \\geq \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B;\n\t\t\\end{equation}\n\t\t\\item Case (ii)\n\t\t\\begin{equation}\\label{case2.2}\n\t\t\t\\| g \\|_A \\leq \\gamma \\| \\partial_{\\nu}u \\|_A, \\quad \\delta \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A \\leq \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B.\n\t\t\\end{equation}\n\t \\item Case (iii)\n\t \\begin{equation}\\label{case1.1}\n\t \t\\| g \\|_A \\geq \\gamma \\| \\partial_{\\nu}u \\|_A;\n\t \\end{equation}\n\t\\end{itemize}\n\tThe constant $\\gamma$ and $\\delta$ are small numbers which will be chosen later.\n\t\n\t\n\tFor Case (i),\n\tit yields from \\eqref{case2Re} that\n\t\\begin{equation}\\label{case2.1.1}\n\t\t\\begin{aligned}\n\t\t&\\delta \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 + \\gamma \\pi \\left\\| \\left( \\partial_{\\nu}u \\right)_n \\right\\|_A \\|u\\|_B \\\\\n\t\t&\\geq \\left(\\sum_{n \\in \\mathbf{H}} \\left( \\partial_{\\nu}u \\right)_n \\bar{u}_n + \\sum_{n \\in \\mathbf{H}}\\left(\\partial_{\\nu} v\\right)_{n}\\overline{v_{n}} \\right)\n\t\t+\\Lambda \\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tFor the second term in \\eqref{case2.1.1}, Case (i) and Lemma \\ref{Lem:H} yield\n\t\\begin{equation}\\label{case2.1.2}\n\t\t\\begin{aligned}\n\t\t\t\\gamma \\pi \\left\\| \\left( \\partial_{\\nu}u \\right)_n \\right\\|_A \\|u\\|_B\n\t\t\t\\leq & \\gamma \\pi \\left( \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A + C \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B \\right) \\times \\left( \\delta \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A + C \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B \\right)\\\\\n\t\t\t\\leq & C' \\left( \\gamma \\delta \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 + \\gamma^2 \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 + \\frac{1}{2} \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 \\right).\n\t\t\\end{aligned}\n\t\\end{equation}\n\tSince\n\t\\begin{equation}\\label{case2.1frac12}\n\t\t\\gamma^2 \\| \\partial_{\\nu}u \\|_A^2\n\t\t\\geq \\| \\partial_{\\nu}u -T(u) \\|_A^2\n\t\t\\geq \\left\\| \\mathcal{P}_\\mathbf{L} - \\mathcal{T}_\\mathbf{L} \\right\\|_A^2\n\t\t\\geq \\frac{1}{2} \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 - \\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2,\n\t\\end{equation}\n\tone has\n\t\\begin{equation}\\label{case2.1.3}\n\t\t\\frac{1}{2} \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 \\leq \\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2 + \\gamma^2 \\| \\partial_{\\nu}u \\|_A^2\n\t\t\\leq \\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2 + \\gamma^2 \\left( \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 + C \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 \\right).\n\t\\end{equation}\n\tBy \\eqref{case2.1.1}, \\eqref{case2.1.2} and \\eqref{case2.1.3},\n\tchoosing\n\t\\begin{equation}\\label{gamma}\n\t\t\\gamma=C_{(i)}\\sqrt{\\delta},\n\t\\end{equation}\n\twhere $C_{(i)}$ is an arbitrarily sufficiently small constant independent of $k$ and $b$, it can be deduced that\n\t\\begin{equation}\\label{Tu_L}\n\t\t\\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2 \\geq \\dfrac{C_1}{\\delta} \\left( \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + \\Lambda \\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2 \\right).\n\t\\end{equation}\n\t\n\n\tIn order to get a bound for $\\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2$, it is sufficient to estimate $(Tu)_n$ and $(Tv)_n$ for $n\\leq k$ due to the definition of $\\mathbf{L}$.\n\tBy the Cauchy-Schwarz inequality, one has\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t& \\sum_{\\vert n \\vertk} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2 \\\\\n\t\t\t\\geq & \\lvert \\sum_{n\\in \\mathbb{Z} } \\sqrt{{k} ^{2} -n^2} \\hat{u} _n \\overline{\\widehat{\\sin(n x_1)} } \\rvert^2 \\left( \\sum_{n\\in \\mathbb{Z} } \\sqrt{\\vert{k} ^{2} -n^2\\vert} \\lvert \\widehat{\\sin(n x_1)} \\rvert^2 \\right)^{-1} \\\\\n\t\t\t\\geq & \\lvert (Tu)_n \\rvert^2 \\left( \\sum_{n\\in \\mathbb{Z} } \\sqrt{\\vert{k} ^{2} -n^2\\vert} \\lvert \\widehat{\\sin(n x_1)} \\rvert^2 \\right)^{-1} \\\\\n\t\t\t\\geq & \\frac{1} {2k} \\lvert (Tu)_n \\rvert^2 \\quad n < k,\n\t\t\\end{aligned}\n\t\\end{equation}\n\tand\n\t\\begin{equation} \\label{CSv}\n\t\t\\begin{aligned}\n\t\t\t& \\sum_{\\vert n \\vertk} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2 \\\\\n\t\t\t\\geq & \\lvert \\sum_{n\\in \\mathbb{Z} } \\sqrt{{k} ^{2} -n^2} \\hat{u} _n \\overline{\\widehat{\\cos(n x_1)} } \\rvert^2 \\left( \\sum_{n\\in \\mathbb{Z} } \\sqrt{\\vert{k} ^{2} -n^2\\vert} \\lvert \\widehat{\\cos(n x_1)} \\rvert^2 \\right)^{-1} \\\\\n\t\t\t\\geq & \\lvert (Tv)_n \\rvert^2 \\left( \\sum_{n\\in \\mathbb{Z} } \\sqrt{\\vert{k} ^{2} -n^2\\vert} \\lvert \\widehat{\\cos(n x_1)} \\rvert^2 \\right)^{-1} \\\\\n\t\t\t\\geq & \\frac{1} {2k} \\lvert (Tv)_n \\rvert^2 \\quad n < k.\n\t\t\\end{aligned}\n\t\\end{equation}\n\t\n\tNote that\n\t\\begin{equation}\\label{LCy0}\n\t\\begin{aligned}\n\t\t\\sum_{n\\in\\mathbf{L}} \\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}} &= \\sum_{n=0}^{\\lfloor k \\rfloor-1}\\frac{1}{\\sqrt{\\lvert k^{2}-n^{2}\\rvert}+\\frac{1}{b}} + \\frac{1}{\\sqrt{\\lvert k^{2}-\\lfloor k \\rfloor^2 \\rvert}+\\frac{1}{b}} \\\\\n\t\t&\\leq \\displaystyle\\int_0^{\\lfloor k \\rfloor} \\frac{1}{\\sqrt{\\lvert k^{2}-x^{2}\\rvert}+\\frac{1}{b}} \\mbox{d}x + b \\\\\n\t\t&\\leq C + b \\leq {C_2}b\n\t\\end{aligned}\t\n\t\\end{equation}\n\tsince $b>1$.\n\tA combination of the above estimate \\eqref{LCy0} with \\eqref{Tu_L}-\\eqref{CSv} yields\n\t\\begin{equation} \\label{case2.1leqgeq}\n\t\t\\begin{aligned}\n\t\t\t& C_2 b k \\left( \\Lambda \\sum_{\\vert n \\vertk} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2 \\right) \\\\\n\t\t\t\\geq & \\sum_{n\\in\\mathbf{L} } \\frac{1} {\\sqrt{\\lvert {k} ^{2} -n^{2} \\rvert} +\\frac{1} {b} } \\left( \\lvert (Tu)_n \\rvert^2 + \\lvert (Tv)_n \\rvert^2 \\right) \\\\\n\t\t\t= & \\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2 \\\\\n\t\t\t\\geq & \\frac{C_1} {\\delta} \\left( \\left\\|\\mathcal{V}_\\mathbf{H}\\right\\|_B^2 + \\Lambda \\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2 \\right) \\\\\n\t\t\t\\geq & \\frac{C_1} {\\delta} \\Lambda \\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tBy choosing\n\t\\begin{equation}\\label{delta}\n\t\t\\delta = \\frac{C_{(ii)}}{ b k},\n\t\\end{equation}\n\twith $C_{(ii)} = \\frac{2 C_1}{1 + 2 C_2}>0$, which is independent of $k$ and $\\varepsilon$, one has\n\t\\begin{equation}\\label{ineq:Cnk}\n\t\tC \\sum_{\\vert n \\vertk} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2.\n\t\\end{equation}\n\tApplying the above estimate \\eqref{ineq:Cnk} and the choice of $\\delta$ \\eqref{delta} to \\eqref{case2.1leqgeq}, one has\n\t\\begin{equation}\\label{ineq:bk}\n\t\tC \\sum_{\\vert n \\vertk} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2.\n\t\\end{equation}\n\tBy combining \\eqref{case2Im}, \\eqref{ineq:bk}, Lemma \\ref{Lem:sqrtk} and the proposition \\eqref{ineq:normB} of norm $B$, one gets\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tC k \\lvert \\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1\\rvert\n\t\t\n\t\t\t\\geq & k \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + C'' \\sqrt{\\varepsilon} k \\sqrt{k} \\|u\\|_{L^2(\\Gamma)} ^2\n\t\t\t\\geq k \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + \\sqrt{\\varepsilon} \\sqrt{k} \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2\n\t\t\t\\geq \\sqrt{\\varepsilon} \\sqrt{k} \\|u\\|_B^2.\n\t\t\\end{aligned}\n\t\\end{equation*}\n\tHence,\n\t\\begin{equation}\\label{case21uB}\n\t\t\\dfrac{ C \\sqrt{k} } {\\sqrt{\\varepsilon} } \\|g\\|_A \\geq \\|u\\|_B.\n\t\\end{equation}\n\t\n\tMoreover, since Lemma \\ref{Lem:H} indicates\n\t\\begin{equation}\\label{case2.1.H}\n\t\t\\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 \\geq C \\left\\| \\mathcal{P}_\\mathbf{H} \\right\\|_A^2,\n\t\\end{equation}\n\tand \\eqref{case2.1frac12} implies\n\t\\begin{equation}\\label{case2.1frac12.2}\n\t\t\\left\\| \\mathcal{T}_\\mathbf{L} \\right\\|_A^2 \\geq \\frac{1}{2} \\left\\| \\mathcal{P}_\\mathbf{L} \\right\\|_A^2 - \\gamma^2 \\| \\partial_{\\nu}u \\|_A^2,\n\t\\end{equation}\n\ta combination of the above \\eqref{case2.1.H}-\\eqref{case2.1frac12.2} with \\eqref{case2Im}, \\eqref{case2.1leqgeq} and Lemma \\ref{Lem:H} yields\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tC b k \\pi \\|g\\|_A \\|u\\|_B\n\t\t\t\\geq C b k \\lvert \\int_{\\Gamma} g \\bar{u} \\mathrm{d} x_1\\rvert\n\t\t\t\\geq C b k \\Lambda \\sum_{\\vert n \\vertk} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2.\n\t\\end{equation}\n\tFor the second term in \\eqref{case2.2.0.1}, Case (ii) and Lemma \\ref{Lem:H} yield\n\t\\begin{equation}\\label{case2.2.0.2}\n\t\t\\begin{aligned}\n\t\t\t\\gamma\\pi \\|\\partial_{\\nu}u \\|_A \\|u\\|_B\n\t\t\t\\leq & \\gamma\\pi \\left( \\frac{1}{\\delta} \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B + C \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B \\right) \\times \\left( \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B + \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B \\right) \\\\\n\t\t\t\\leq & C \\frac{\\gamma}{\\delta} \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2 + C \\gamma^2 \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2 + \\frac{1}{2}\\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tSince $\\gamma=C_{(i)}\\sqrt{\\delta}$ is sufficiently small, the estimate \\eqref{case2.2.0.1} and \\eqref{case2.2.0.2} indicate that\n\t\\begin{equation}\\label{case2.2.0}\n\t\t\\frac{C}{\\delta} \\left\\| \\mathcal{V}_\\mathbf{L} \\right\\|_B^2\n\t\t\\geq \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2 + \\Lambda \\sum_{\\vert n \\vert>k} \\sqrt{n^2-{k} ^{2} } \\vert \\hat{u} _n \\vert^2 \\geq \\left\\| \\mathcal{V}_\\mathbf{H} \\right\\|_B^2.\n\t\\end{equation}\n\tOn the other hand, for the lower frequency, it is obvious that\n\t\\begin{equation*}\n\t\tC \\Lambda \\sum_{\\vert n \\vert1$ and large $k$, such that\n\t\\begin{equation*}\n\t\t\\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\leq C \\max \\left\\{ \\dfrac{ b^{\\frac{1} {2} } k } {\\sqrt{\\varepsilon} }\n\t\t, b^{\\frac{3} {2} } {k} ^{\\frac{3} {2} } \\right\\} \\|g\\|_A .\n\t\\end{equation*}\n\tFurthermore, Lemma \\ref{lem:step1} gives\n\t\\begin{equation*}\n\t\t\\|\\nabla u \\|_{L^2(D)} + k \\|u\\|_{L^2(D)} \\leq C b k \\left( \\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\right) .\n\t\\end{equation*}\n\tTherefore,\n\t\\begin{equation}\\label{eq:2.1}\n\t\t\\| \\partial_{\\nu}u \\|_A + \\|u\\|_B \\leq C \\max \\left\\{ \\dfrac{ b^{\\frac{3} {2} } {k} ^2 } {\\sqrt{\\varepsilon} }\n\t\t, b^{\\frac{5} {2} } {k} ^{\\frac{5} {2} } \\right\\} \\|g\\|_A .\n\t\\end{equation}\n\tFrom the definition and the property of the norm $A$ mentioned in Definition \\ref{def:AB} and Remark \\ref{rem:AL},\n\t\\begin{equation}\\label{eq:2.2}\n\t\t\\|g\\|_A \\leq C \\sqrt{b} \\|g\\|_{L^2(\\Gamma)} .\n\t\\end{equation}\n\t\n\tApplying \\eqref{eq:2.2} to \\eqref{eq:2.1}, one derives the bounds in Theorem \\ref{thm:1}, which completes the proof.\n\\end{proof}\n\n\n\\section{Stability for the random case}\n\\label{sec:thm23}\n\nIn this section, Theorem \\ref{thm:3} is proved by utilizing the general framework developed in \\cite{Pembery2020} based upon the stability in Theorem \\ref{thm:1} for the deterministic case.\nNote that the randomness of integral domain prevents direct application of general framework in \\cite{Pembery2020}.\nThus the difficulty lies in that the scattering surface in model problem is\nwith randomness, which implies the stochastic domain for the scattering problem.\nFor this, a variable transform, which changes the random variational form with stochastic domains into a transformed one with a definite domain and random medium, is introduced to prove all necessary propositions, including the continuity of sesquilinear form $\\tilde{a}$ and antilinear functional $\\tilde{G}$, and stability and uniqueness which hold almost surely, which implies the stability for the random case.\n\n\n\\subsection{Prerequisites}\n\\label{sec:4.preliminary}\n\nDenote $\\tilde{a}_{c(\\omega)}$ and $\\tilde{G}_{c(\\omega)}$ (defined by \\eqref{aw} and \\eqref{Gw} in Sec \\ref{sec:variational}) by $(\\tilde{a} \\circ c)(\\omega)$ and $(\\tilde{G} \\circ c)(\\omega)$, respectively.\nDefine the norm $\\|v\\|^2_{1,k} := \\|\\nabla v \\|_{L^2(D)}^2 + k^2 \\|v\\|_{L^2(D)}^2$ and $\\|v\\|_{1,\\infty}:=\\|v\\|_{\\infty}+\\|v'\\|_{\\infty}$ on $H_{S,qp}^1(D)$.\nNecessary propositions required in the general framework \\cite{Pembery2020} include the continuity of $\\tilde{a}$ and $\\tilde{G}$, regularity of $\\tilde{a}\\circ c$ and $\\tilde{G}\\circ c$, measurability and $\\mu$-essentially separability of $c$, stability a.s. and uniqueness a.s., described in Proposition \\ref{prop:1} and Proposition \\ref{prop:2}.\n\n\\begin{proposition}\n\t\\label{prop:1}\n\t$\\tilde{a}$, $\\tilde{G}$ and $c$ in the variational form \\eqref{variational} of the scattering problem in a random periodic structure satisfies the following properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] The function $c: \\Omega \\rightarrow \\mathcal{C}$ defined by \\eqref{def:c} is measurable and $\\mu$-essentially separably valued.\n\t\t\\item[(ii)]\n\t\tLet $B(H_{S,qp}^1(D), H_{S,qp}^1(D))$ denote the space of bounded linear maps $H_{S,qp}^1(D) \\rightarrow H_{S,qp}^1(D)$. The functions $\\tilde{a}: \\mathcal{C}\\rightarrow B(H_{S,qp}^1(D),H_{S,qp}^1(D))$ and $\\tilde{G}:\\mathcal{C}\\rightarrow H_{S,qp}^1(D)$ defined by \\eqref{aw} and \\eqref{Gw} are continuous,\n\t\tthe maps $\\tilde{a}\\circ c\\in L^{\\infty}(\\Omega;B(H_{S,qp}^1(D),H_{S,qp}^1(D)))$, $\\tilde{G}\\circ c\\in L^2(\\Omega;H_{S,qp}^1(D))$.\n\t\n\t\\end{itemize}\n\\end{proposition}\n\n\n\\begin{definition}\n\t(The solution operator $\\mathcal{U}$)\n\tDefine $\\mathcal{U}:\\mathcal{C}\\rightarrow H_{S,qp}^1(D)$ by letting $\\mathcal{U}(f_0) \\in H_{S,qp}^1(D)$ be the solution of the deterministic Helmholtz problem with sampling $f_0\\in\\mathcal{C}$.\n\\end{definition}\n\n\n\\begin{proposition} \\label{prop:2}\n\tFor $f_0\\in Lip$, the solution of the variational form \\eqref{variational} exists and is unique.\n\tLet $v_0=\\mathcal{U}(f_0)$ be the solution. Under Assumption \\ref{assump:k}, there exists a constant $C$ independent of $b>1$ and large $k$, such that\n\t\\begin{equation}\\label{Destimatelem}\n\t\t\\|\\nabla v_0 \\|_{L^2(D)} + k \\|v_0\\|_{L^2(D)} \\leq C \\max \\left\\{ \\dfrac{b^2 {k} ^{2} } {\\sqrt{\\varepsilon} } ,b^3 {k} ^{\\frac{5} {2} } \\right\\} \\|g\\|_{L^2(\\Gamma)}.\n\t\\end{equation}\n\tMoreover, the scattering problem \\eqref{eq:u_total} in a random periodic structure satisfies the following stability and uniqueness properties:\n\t\\begin{itemize}\n\t\t\\item[(i)] (Uniqueness almost surely) $ker(\\tilde{a}_{c(\\omega)})=\\{0\\}$ $\\mu$-almost surely.\n\t\t\\item[(ii)] (Stability almost surely) There exist $C_1, h_1:\\Omega\\rightarrow\\mathbb{R}$ such that $C_1 h_1\\in L^1(\\Omega)$ and the bound\n\t\t\\begin{equation}\n\t\t\t\\|u(\\omega)\\|_{1,k}^2\\leq C_1(\\omega) h_1(\\omega)\n\t\t\\end{equation}\n\t\tholds almost surely.\n\t\\end{itemize}\n\\end{proposition}\n\n\\begin{remark}\n\tFrom the general framework in \\cite{Pembery2020}, one can conclude that (i) with properties of $\\tilde{a}$, $\\tilde{G}$ and $c$ stated in Proposition \\ref{prop:1}, the maps $\\mathcal{A}$ and $\\mathcal{G}$ defined by \\eqref{def:AA} and \\eqref{def:GG} are well-defined;\n\t(ii) with the almost surely uniqueness property in Proposition \\ref{prop:2}, the solution to the Helmholtz equation in the stochastic case is unique in $L^2(\\Omega;H_{S,qp}^1(D))$;\n\t(iii) with the almost surely stability property in Proposition \\ref{prop:2}, integrability and measurability implies the stochastic a priori bound.\n\\end{remark}\n\nIn order to verify Proposition \\ref{prop:1} and Proposition \\ref{prop:2}, we give the following four lemmas. Lemma \\ref{lem:Kirsch} means that the solution to the scattering problem depends continuously on the scattering surface.\nLemma \\ref{lem:Pettis} shows the equivalence between strong measurability and measurability as well as $\\mu$-essentially separability.\nLemma \\ref{lem:Bochner} shows the necessary and sufficient condition for Bochner integrability.\nLemma \\ref{u_welldefined} shows that the solution operator $\\mathcal{U}$ is well defined.\n\n\\begin{lemma}\\label{lem:Kirsch}\n\t(continuous dependence on the boundary curve $f$ \\cite{Kirsch1993}) Let $f\\in C^2(\\mathbb{R})$ be $2 \\pi$-periodic and $u\\in H_{S,qp}^1$ be the unique solution of the scattering problem \\eqref{eq:u_deter}. Let $K \\subset D$ be compact. Then there exists $\\gamma>0$ and $C>0$ both depending on $k, u^i, f$ and $K$, such that for all $2\\pi$-periodic $r\\in C^1(\\mathbb{R})$ with $\\|r-f\\|_{1,\\infty}\\leq\\gamma$, the unique solution $u_r$ of the scattering problem corresponding to $r$ satisfies\n\t$$ \\|u_r-u\\|_{H^1(K)} \\leq C \\|f-r\\|_{1,\\infty}.$$\n\tHere, $\\|q\\|_{1,\\infty}:=\\|q\\|_{\\infty}+\\|q'\\|_{\\infty}$ denotes the norm in $C^1[0,\\Lambda]$, where $\\Lambda$ is the period of the scattered structures.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:Pettis}\n\t(Pettis measurability theorem\n\t(Proposition 2.15 in \\cite{Ryan2002})). Let $(\\Omega, \\mathcal{F}, \\mu)$ be a complete $\\sigma$-finite measure space. The following are equivalent for a function $f: \\Omega \\rightarrow X$ (i) $f$ is strongly measurable, (ii) $f$ is measurable and $\\mu$-essentially separably valued.\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:Bochner}\n\t(Bochner's Theorem (Proposition 2.16 in \\cite{Ryan2002})). If $f: \\Omega \\rightarrow X$ is a strongly measurable function, then $f$ is Bochner integrable if and only if the scalar function $\\|f\\|$ is integrable.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{u_welldefined}\n\t( $\\mathcal{U}$ is well defined)\n\tFor $f_0\\in\\mathcal{C}$, the solution $\\mathcal{U}$ of the scattering problem exists, is unique, and depends continuously on $f_0$.\n\\end{lemma}\n\n\\begin{proof}\t\n\tFor $f_0\\in\\mathcal{C}$, the scattering problem is a deterministic case satisfying the condition in Theorem \\ref{thm:1}. Based on the previous work in \\cite{Bao1995} on the existence and uniqueness, it follows from Theorem \\ref{thm:1} and Lemma \\ref{lem:Kirsch} that the solution $\\mathcal{U}$ is well defined.\n\\end{proof}\n\nHere the basic propositions on measure theory and Bochner spaces needed for the proof are omitted. See \\cite{Bogachev2007} and \\cite{Diestel1977} for more details.\n\n\n\\subsection{Proof of Proposition \\ref{prop:1} and Proposition \\ref{prop:2}}\\label{sec:4.verification}\n\nIn this section, we verify that the random periodic structure scattering problem has all necessary propositions required as in the general framework \\cite{Pembery2020}.\nProposition \\ref{prop:1} includes\nmeasurability and\n$\\mu$-essentially separability of $c$, continuity of $\\tilde{a} $ and $\\tilde{G} $ and\nregularity of $\\tilde{a} \\circ c$ and $\\tilde{G} \\circ c$.\nProposition \\ref{prop:2} includes the necessary\nstability and uniqueness which hold almost surely. To be specific, under the assumpiton of resonance exclusion and stochastic regularity of the scattering surface, a variable transform is introduced and theory of Bochner spaces such as Pettis measurability theorm and Bochner's theorem is used to complete the proof.\n\nFirst, we give the proof of Proposition \\ref{prop:1}.\n\n\\begin{proof}[Proof of Proposition~{\\upshape\\ref{prop:1}}]\n\n\tSince each of $f$ is a Lipschitz function, $f$ is strongly measurable. By Pettis measurability theorem (Lemma \\ref{lem:Pettis}), it follows that $c$ defined by Definition \\ref{def:c} is measurable and $\\mu$-essentially separably valued, so properties of $c$ in Proposition \\ref{prop:1} are satisfied.\n\t\n\n\tIn order to prove the continuity of $\\tilde{a}$ and $\\tilde{G}$, we need to show that if $(f_m)\\rightarrow(f_0)$ in $\\mathcal{C}$ then $\\tilde{a}(f_m)\\rightarrow \\tilde{a}(f_0)$ in $B(H_{S,qp}^1(D),H_{S,qp}^1(D))$, and similarly for $\\tilde{G}$. Since\n\t\\begin{equation}\\label{varia:f_m}\n\t\t\\tilde{a}_{f_m}(v_m,\\phi)=\\tilde{G}_{f_m}(\\phi),\n\t\\end{equation}\n\twhere\n\t\\begin{equation}\n\t\t\\tilde{a}_{f_m} =\\int_{D_{f_m}} \\nabla v_m \\cdot \\nabla \\bar{\\phi}\n\t\t- k^2 \\int_{D_{f_m}} v_m \\bar{\\phi}\n\t\t- \\int_{\\Gamma} T \\tau v_m \\overline{\\tau \\phi} , \\quad \\tilde{G}_{f_m} = \\int_{\\Gamma} g \\overline{\\tau \\phi} ,\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{varia:f_0}\n\t\t\\tilde{a}_{f_0}(v_0,\\phi)=\\tilde{G}_{f_0}(\\phi),\n\t\\end{equation}\n\twhere\n\t\\begin{equation}\\label{varia:f_0aG}\n\t\t\\tilde{a}_{f_0} =\\int_{D_{f_0}} \\nabla v_0 \\cdot \\nabla \\bar{\\phi}\n\t\t- k^2 \\int_{D_{f_0}} v_0 \\bar{\\phi}\n\t\t- \\int_{\\Gamma} T \\tau v_0 \\overline{\\tau \\phi} , \\quad \\tilde{G}_{f_0} = \\int_{\\Gamma} g \\overline{\\tau \\phi}.\n\t\\end{equation}\n\tOur goal is to transform the volume integral in $D_{f_m}$ into an integral over $D_{f_0}$ by a suitable change of the variable $x$.\n\t\n\n\tChoose $\\gamma_0>0$ such that\n\t\\begin{equation}\\label{def:gamma}\n\t\t\\gamma_0<\\min\\{\\min\\{x_2:(x_1,x_2)\\in D_{f_0}\\}-f_0(x_1):x_1\\in\\mathbb{R}\\},\n\t\\end{equation} and a function $\\alpha\\in C^1(\\mathbb{R})$ with\n\t\\begin{equation}\\label{def:alpha}\n\t\t\\alpha(t) =\\left\\{ \\begin{array}{ll}\n\t\t\t1, & t \\leq \\gamma_0\/2 \\\\\n\t\t\t0, & t\\geq \\gamma_0.\n\t\t\\end{array} \\right..\n\t\\end{equation}\t\n\tFor $f_m$ such that $\\|f_m-f_0\\|_{1,\\infty} \\leq \\gamma_0$, define $\\mathcal{F}: D_{f_0} \\rightarrow D_{f_m}$ by\n\t$$\n\t\\mathcal{F}(y) = y_2 + \\alpha(y_2-f_0(y_1)) [f_m(y_1)-f_0(y_1)] \\hat{e}_2,\\quad y\\in D_{f_0},\n\t$$\n\twhere $\\hat{e}_2=(0,1)^T$.\t\n\tFor $\\|f_m-f_0\\|_{1,\\infty} \\leq \\gamma$ with sufficiently small $\\gamma\\leq \\gamma_0$, $\\mathcal{F}$ is a diffeomorphism and $\\mathcal{F}=\\mathcal{I}$ on $K$. The Jacobian $J_{\\mathcal{F}}$ of $\\mathcal{F}$ satisfies $J_{\\mathcal{F}}(y) = \\mathcal{I} + \\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty})$ uniformly in $y\\in D_{f_0}$. For the inverse $\\mathcal{Q}$ of $\\mathcal{F}$, one also has $J_{\\mathcal{Q}}(y) = \\mathcal{I} + \\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty})$. The change of variable $x=\\mathcal{F}(y)$ then transforms \\eqref{varia:f_m} into\n\t\\begin{equation}\\label{eq:varia_new}\n\t\t\\int_{D_{f_0}}\\left[\\sum_{i, j=1}^{2} b_{i j} \\frac{\\partial \\hat{v}_{m}}{\\partial y_{i}} \\frac{\\partial \\overline{\\hat{\\phi}}}{\\partial y_{j}}-k^{2} \\hat{v}_{m} \\overline{\\hat{\\phi}}\\right] \\operatorname{det} (J_{\\mathcal{F}}) \\mathrm{d} y-\\int_{\\Gamma} \\overline{\\tau \\hat{\\phi}} T \\tau \\hat{v}_{m} \\mathrm{d} s=\\int_{\\Gamma} g \\overline{\\tau \\hat{\\phi}} \\mathrm{d} s,\n\t\\end{equation}\n\twhere $\\hat{v}_{m}=v_{m} \\circ \\mathcal{F}, \\hat{\\phi}=\\phi \\circ \\mathcal{F}$ both in $H_{S,qp}^1(D_{f_0})$, and\n\t\\begin{equation}\\label{bij}\n\t\tb_{i j}(y)=\\sum_{l=1}^{2} \\frac{\\partial \\mathcal{Q}_{i}(x)}{\\partial x_{l}} \\frac{\\partial \\mathcal{Q}_{j}(x)}{\\partial x_{l}}\\rvert_{x=\\mathcal{F}(y)}, \\quad i, j = 1, 2.\n\t\\end{equation}\n\tThe left hand side of \\eqref{eq:varia_new} defines a sesquilinear form $\\tilde{a}_{f_m}(v,\\phi)$ on $H_{S,qp}^1(D_{f_0})$.\n\tSince $ \\operatorname{det} (J_{\\mathcal{F}}) = 1 + \\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty}) $ and $ b_{ij}=\\delta_{ij}+\\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty}) $, one concludes that\n\t$$\\lvert \\tilde{a}_{f_m}(v, \\phi)-\\tilde{a}_{f_0}(v, \\phi)\\rvert \\leq C\\|f_m-f_0\\|_{1, \\infty}\\|v\\|_{H^{1}\\left(D_{f_0}\\right)}\\|\\phi\\|_{H^{1}\\left(D_{f_0}\\right)} \\text { for all } v, \\phi \\in H_{S,qp}^1(D_{f_0}).$$\n\tHence if $(f_m)\\rightarrow(f_0)$ in $\\mathcal{C}$, then $\\tilde{a}_{f_m}\\rightarrow \\tilde{a}_{f_0}$ in $B(H_{S,qp}^1(D),H_{S,qp}^1(D))$. For $\\tilde{G}$, it follows from the definition \\eqref{varia:f_0}-\\eqref{varia:f_0aG}, the right hand side of \\eqref{eq:varia_new}, $\\hat{\\phi}=\\phi \\circ \\mathcal{F} $ and $\\phi=\\phi \\circ \\mathcal{I}$ that\n\tif $(f_m)\\rightarrow(f_0)$ in $\\mathcal{C}$, then $\\tilde{G}_{f_m}\\rightarrow \\tilde{G}_{f_0}$ in $H_{S,qp}^1(D)$.\n\t\n\n\tSince $c$ is strongly measurable and the map $\\tilde{a}$ is continuous, $\\tilde{a}\\circ c $ is strongly measurable. Recall that the operator $T$ is continuous from $H^{1\/2}(\\Gamma)$ to $H^{-1\/2}(\\Gamma)$ \\cite{Nedelec2001}. For $v\\in H_{S,qp}^1(D)$, $\\phi\\in H_{S,qp}^1(D)$, one has\n\t\\begin{equation}\\label{eq:afomega}\n\t\t\\tilde{a}_{f(\\omega)}(v, \\phi) =\\int_{D_{f(\\omega)}} \\nabla v \\cdot \\nabla \\bar{\\phi}\n\t\t- k^2 \\int_{D_{f(\\omega)}} v \\bar{\\phi}\n\t\t- \\int_{\\Gamma} T \\tau v \\overline {\\tau \\phi}.\n\t\\end{equation}\n\tSame as above, transform the volume integral in $D_{f(\\omega)}$ into an integral over $D_{f_0}$.\n\tFor $f_1=f(\\omega)$ such that $\\|f_1-f_0\\|_{1,\\infty} \\leq \\gamma_0$ (defined same as \\eqref{def:gamma}), define $\\mathcal{F}: D_{f_0} \\rightarrow D_{f_1}$ by\n\t$$\n\t\\mathcal{F}(y) = y_2 + \\alpha(y_2-f_0(y_1)) [f_1(y_1)-f_0(y_1)] \\hat{e}_2,\\quad y\\in D_{f_0},\n\t$$\n\twhere $\\hat{e}_2=(0,1)^T$ and $\\alpha$ is defined by \\eqref{def:alpha}.\t\n\tFor $\\|f_1-f_0\\|_{1,\\infty} \\leq \\gamma$ with sufficiently small $\\gamma \\leq \\gamma_0$, $\\mathcal{F}$ is a diffeomorphism and $\\mathcal{F}=\\mathcal{I}$ on $K$. The change of variable $x=\\mathcal{F}(y)$ then transforms \\eqref{eq:afomega} into\n\t\\begin{equation}\n\t\t\\tilde{a}_{f(\\omega)}(\\hat{v}, \\phi)=\\int_{D}\\left[\\nabla \\hat{v}\\operatorname{det} (J_{\\mathcal{F}}) B \\nabla \\bar{\\phi}-k^{2}(\\operatorname{det} (J_{\\mathcal{F}}) \\hat{v} \\bar{\\phi}\\right] d y,\n\t\\end{equation}\n\twhere $\\hat{v}=v \\circ \\mathcal{F}, \\hat{\\phi}=\\phi \\circ \\mathcal{F}$ both in $H_{S,qp}^1(D_{f_0})$, and $B=(b_{ij})_{i,j=1,2}$ from \\eqref{bij}. \t\t\n\tSince $ \\operatorname{det} (J_{\\mathcal{F}}) = 1 + \\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty}) $ and $ b_{ij}=\\delta_{ij}+\\mathcal{O}(\\|f_m-f_0\\|_{1,\\infty}) $, observe that the Cauchy-Schwarz inequality and properties of $T$ imply that there exists $C>0$ such that\n\t$$\n\t\\lvert \\tilde{a}_{f(\\omega)}(\\hat{v}, \\phi)\\rvert \\leq\n\tC \\|f(\\omega)-f_0\\|_{1,\\infty}^2 \\|v\\|_{H^1(D)} \\|\\phi\\|_{H^1(D)} \\text { for all } v, \\phi \\in H_{S,qp}^1(D_{f_0}),\n\t$$\n\tand hence $\\tilde{a}\\circ c \\in L^{\\infty}(\\Omega;B(H_{S,qp}^1(D),H_{S,qp}^1(D))) $.\n\t\n\n\tSince $c$ is strongly measurable and the map $\\tilde{G}$ is continuous, $\\tilde{G}\\circ c $ is strongly measurable. It is clear that $\\| (\\tilde{G}\\circ c)(\\omega) \\|_{H_{S,qp}^1(D)} \\leq \\|g\\|_{L^2(D)}$, and thus $\\tilde{G}\\circ c \\in L^2(\\Omega;H_{S,qp}^1(D))$ since $g \\in L^2(\\Omega;L^2(D))$.\n\\end{proof}\n\nNow prove Proposition \\ref{prop:2}.\n\n\\begin{proof}[Proof of Proposition~{\\upshape\\ref{prop:2}}]\n\tFor $f_0\\in \\mathcal{C}$, it follows from Theorem \\ref{thm:1} and Lemma \\ref{lem:Kirsch} that the solution $\\mathcal{U}(f_0)$ of the variational problem exists, is unique, and has the $k$-explicit stability estimate \\eqref{Destimatelem}. \t\n\tUniqueness almost surely holds immediately from Lemma \\ref{u_welldefined}.\n\t\n\tFor stability almost surely, choose\n\t$C_1 =C' \\max \\left\\{ \\dfrac{b^2 {k} ^{2} } {\\sqrt{\\varepsilon} } ,b^3 {k} ^{\\frac{5} {2} } \\right\\} $ with $C'=\\sup_\\Omega {C}$, $C$ being the previous constant in \\eqref{Destimatelem}, and $h_1 = \\|g\\|_{L^2(\\Gamma)}$.\n\tIt remains to show that $C_1 h_1 \\in L^1(\\Omega)$.\n\tWe first show that $C_1 h_1$ is measurable and then show that it lies in $ L^1(\\Omega)$.\n\t\n\tTo conclude $C_1$ is measurable, use the fact that the product, sum and maximum of two measurable functions are measurable. Since $g$ is measurable and the map $g\\mapsto \\|g\\|_{L^2(\\Gamma)}^2$ is clearly continuous, $h_1$ is measurable. As the product of two measurable functions is measurable, it follows that $C_1 h_1$ is measurable.\n\t\n\tNow show that $C_1 h_1 \\in L^1(\\Omega)$. Using the Cauchy-Schwarz inequality yields\n\t\\begin{eqnarray} \\label{C1h1L1}\n\t\t\\left\\|C_{1}h_1\\right\\|_{L^{1}(\\Omega)}=\\int_{\\Omega} C_{1}(\\omega)h_1 (\\omega) \\mathrm{~d} \\mathbb{P}(\\omega)\\leq \\left\\|C_{1}\\right\\|_{L^{1}(\\Omega)}\\left\\|\\left\\|g\\right\\|_{L^{2}\\left(\\Gamma\\right)}^{2}\\right\\|_{L^{1}(\\Omega)}<\\infty.\n\t\\end{eqnarray}\n\tTherefore $C_1 h_1 \\in L^1(\\Omega)$ as required. Integrating \\eqref{Destimatelem} in the probability space and using \\eqref{C1h1L1} obviously yield \\eqref{Sestimate}.\n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{thm:3}}\n\\label{sec:4.proof}\n\nBefore the proof, the continuity of the solution operator $\\mathcal{U}$ is given in the following lemma.\n\n\\begin{lemma}\\label{u_continuity}\n\t(Continuity of $\\mathcal{U}$)\n\tFor the scattering problem by a random periodic structure, the solution operator $\\mathcal{U}:\\mathcal{C}\\rightarrow H_{S,qp}^1(D)$ is continuous.\n\\end{lemma}\n\\begin{proof}\n\tLet $f_0, f_1 \\in \\mathcal{C}$, with $\\mathcal{U}(f_0)=u_0$ and $\\mathcal{U}(f_1)=u_1$. Then for any $v\\in H_{S,qp}^1(D)$,\n\t$$\\tilde{a}_{f_j}(u_j,v)=\\tilde{G}_{f_j}(v),\\ j=0,1.$$\n\t\n\tSince one has\n\t$$\\lvert \\tilde{a}_{f_1}(u, \\phi)-\\tilde{a}_{f_0}(u, \\phi)\\rvert \\leq C\\|f_1-f_0\\|_{1, \\infty}\\|u\\|_{H^{1}\\left(D_{f_0}\\right)}\\|\\phi\\|_{H^{1}\\left(D_{f_0}\\right)} \\text { for all } u, \\phi \\in H_{S,qp}^1(D_{f_0})$$\n\tfrom the proof of Proposition \\ref{prop:1},\n\twhere\n\tboth $\\tilde{a}_{f_1}(u, \\phi)$\n\tand $\\tilde{a}_{f_0}(u, \\phi)$ are sesquilinear forms,\n\tone can applies the general perturbation theory of variational equation \\cite{Kato1976} which yields\n\t$$\\|u_1\\circ\\mathcal{F} - u_0\\|_{H^1(D_{f_0})} = \\|\\hat{u}_1-u_0\\|_{H^1(D_{f_0})}\\leq C \\|f_m-f_0\\|_{1,\\infty},$$\n\tand thus, since $\\mathcal{F}=\\mathcal{I}$ on $D_{f_0}$,\n\t$$\\|u_1-u_0\\|_{H^1(K)} \\leq C \\|f_1-f_0\\|_{1,\\infty}.$$\n\tLet $u_d:=u_0-u_1$. It can be deduced that $u_d\\rightarrow 0$ in $H_{S,qp}^1(D_{f_0})$ as $f_1 \\rightarrow f_0$ in $\\mathcal{C}$. This ends the proof of this lemma.\n\\end{proof}\n\nNow we are ready to prove Theorem \\ref{thm:3}.\n\n\\begin{proof}\n\tLet $u = \\mathcal{U} \\circ c$ (which\n\tis well-defined by Lemma \\ref{u_welldefined}). By construction, $a_{c(\\omega)}(u(\\omega), v)=G_{c(\\omega)}(v)$ for all $v \\in H_{S,qp}^1(D)$ almost surely.\n\tSince it follows from Assumption \\ref{cond:f} and Lemma \\ref{u_continuity} that $u$ is measurable, $u$ solves the variational problem.\n\t\n\tUnder Assumption \\ref{cond:f}, continuity of $\\tilde{a} $ and $\\tilde{G} $,\n\tregularity of $\\tilde{a} \\circ c$ and $\\tilde{G} \\circ c$, and\n\tmeasurability and\n\t$\\mu$-essentially separability of $c$ hold by Proposition \\ref{prop:1};\n\tstability a.s. and uniqueness a.s. hold by Proposition \\ref{prop:2}.\n\tMoreover, since there is no trapping cases for the scattering problem by a Lipschitz perfectly conductor, the nontrapping condition required in \\cite{Pembery2020} is naturally satisfied.\n\tTherefore\n\tthe maps $\\mathcal{A}$ and $\\mathcal{G}$ (defined by \\eqref{def:AA} and \\eqref{def:GG}) are well-defined;\n\tthere exists $u\\in L^2(\\Omega;H^1_{S,qp}(D))$ being the solution to Problem \\ref{P3}; the solution to Problem \\ref{P3} is unique in $L^2(\\Omega;H^1_{S,qp}(D))$.\n\t\n\n\tCombining the stability for the scattering problem by deterministic periodic structures in Theorem \\ref{thm:1} with integrability and measurability properties on stochastic quantities verified by Proposition \\ref{prop:1} and stability and uniqueness which hold almost surely given by Proposition \\ref{prop:2} yields the well-posedness and the $k$-explicit stability for the scattering problem by random periodic structures, which completes the proof of Theorem \\ref{thm:3}.\n\\end{proof}\n\n\\begin{remark}\n\n\tIf the random structure is (uncertainty) quantified using the Karhunen-Lo\\`eve (KL) expansion as in \\cite{BaoLinSINUM2020}, then the random structure can be represented by the following Karhunen-Lo\\`eve expansion\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\tf(\\omega;x_1)&=\\tilde{f}(x_1) + \\displaystyle \\sum_{j=0}^{\\infty}\\sqrt{\\lambda_j} \\xi_{j}(\\omega)\\varphi_{j}(x_1),\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhere $\\tilde{f}(x_1)$ is a $\\Lambda$-periodic deterministic function, eigenvalues $\\{ \\lambda_j \\}_{j=0}^{\\infty}$ arranged in a descending order are corresponding to the orthonormal eigenfunctions $\\{ \\varphi_{j} \\}_{j=0}^{\\infty}$ of the covariance operator $C_f$\n\tand $\\{\\xi_j \\}_{j=0}^{\\infty}$ is a random variable with zero mean and unit covariance. The covariance function \\cite{bookRandomSurface, bookcovariance} takes the following form\n\t\\begin{equation*}\n\t\tc(x_1-y_1)=\\sigma^2 \\exp(-\\dfrac{\\vert x_1-y_1\\vert^2}{l^2}), 00$ whereas the expansion would be smooth when $w_0<0$. In the second one, the value of $\\tilde{w}_2$ is above -1 for negative (positive) values of $w_0$ and $z_0 > -1$ ($z_0 < -1$), so there is no singularity. For positive (negative) values of $w_0$ and $z_0 > -1$ ($z_0 < -1$), the value of $\\tilde{w}_2$ is below $-1$ and a singularity occurs. Thus, depending on the free parameters, the above models may lead to some kind of future singularity. Then, by using Sne Ia data, the best fits are obtained for both models in (\\ref{2.1}) and compared with $\\Lambda$CDM. The results are summarised in the following table, where the best $\\chi^2$ is included as well as the reduced $\\chi^2_{red}$ in order to compare the goodness of the fit for every model. As shown, every model gives a very similar fit, but different cosmological evolutions. Similarly, by using other data sets as BAO, the degeneracy of the results remains, such that parameterisations of the EoS for dark energy are not very useful to find out how the dynamics of dark energy are, at least with the available data sets.\n\\begin{table}[h!] \n\\begin{center}\n\\begin{tabular}{ccccccc}\n\\hline\n\\hline\n\\bf{Model} & $\\bf{\\chi_{min}^2}$ & $\\bf{w_0}$ & $\\bf{z_0}$ & $\\bf{\\Omega_{0m}}$ & $\\bf{\\chi_{red}^2}$\\\\\n\\hline \\vspace{-5pt}\\\\\n$\\Lambda$CDM & $542.685$ & - & - & $0.27 \\pm 0.02$ & $0.978 $\\\\\n\\\\\n$ w_1(z)$ & $542.683$ & $0.0045 \\pm 0.1$ & $-25 \\pm 30$ & $0.27$ & $0.981 $ \\\\\n\\\\\n$ w_2(z)$ & $541.583$ & $-0.03 \\pm 0.07$ & $22 \\pm 45$ & $0.27$ & $0.979 $\\\\\\\\\n \\hline \\hline\n\\end{tabular}\n\\caption{Best fit for the models (\\ref{2.1}) with $\\Omega_{0m}=0.27$ by using the Sne Ia dataset \\cite{union2.1}. The result for the $\\Lambda$CDM model is also shown. \\label{table1}}\n\\end{center}\n\\end{table}\n\\section{Cosmography}\n\\label{cosmography}\nAnother model-independent approach to test cosmology is the so-called cosmography, which is based solely on the cosmological principle regardless of the underlying theoretical model. To do so, the Hubble parameter is expanded in terms of an auxiliary variable\\cite{Weinberg-Harrison}:\n\\begin{equation}\nH(z)=\\frac{\\dot{a}}{a}=H_0+H_{z0}z+\\frac{H_{zz0}}{2}z^2+\\frac{H_{zzz0}}{6}z^3+...\\ ,\n\\label{cosmo1}\n\\end{equation}\nHere we have used the redshift $1+z=\\frac{1}{a}$ as the auxiliary variable and the subscript $0$ refers to quantities evaluated today. Then, the cosmographic parameters are defined in terms of the derivatives of the Hubble parameter, or equivalently in terms of the scale factor as\n\\begin{equation}\nH_0=\\frac{\\dot{a}_0}{a_0}\\ ,\\, q_0=-\\frac{\\ddot{a}_0}{a_0H^2_0}\\ , \\, j_0=\\frac{a_0^{(3)}}{a_0H_0^3}\\ , \\, s_0=\\frac{a_0^{(4)}}{a_0H_0^4}\\ , ...\\ \n\\label{cosmo2}\n\\end{equation}\nwhere the dots are cosmic time derivatives. Then, by inserting (\\ref{cosmo2}) in (\\ref{cosmo1}), the Hubble parameter is written in terms of the cosmographic parameters which should be set with observational data. However, note that the series (\\ref{cosmo1}) converges for $|z|<1$. Hence, alternatively the expansion (\\ref{cosmo1}) may be expressed in terms of another auxiliary variable $y = \\frac{z}{1+z}$ which successes to describe the entire universe history by ensuring the convergence of the series\\cite{Visser}. By generating mock data from a fiducial spatially flat $\\Lambda$CDM model, where we have assumed the same redshifts as the Union 2.1 catalogue \\cite{union2.1}, with errors of magnitude $\\sigma_{\\mu}=0.15$, we can show which variable and order of the series behaves better. We have run 100 simulations and fit the cosmographic parameters by using two different sets of parameters for each variable: $\\bm{\\theta_1} = \\{H_0,q_0,j_0,s_0\\}$ and $\\bm{\\theta_2} = \\{H_0,q_0,j_0,s_0,l_0\\}$, where $H_0$ is marginalised. Then, by using Monte Carlo Markov Chain (MCMC), the corresponding constraints for each set and each variable are obtained and shown in the following table. The table contains the number of times the true parameters were inside the confidence region bounds. It is expected the true value to lie within $1\\sigma$, \n68\\% of the times and $2\\sigma$, 95\\% of the times. The variable $z$ gives well-behaved coverage \nresults for the set $\\theta_1$, overestimates the errors while considering a higher order in the expansion $\\theta_2$. On the other hand, the $y$-parametrisation gives completely biased \nestimators for the set $\\theta_1$, and overestimates the errors for $\\theta_2$. These results show that the variable $z$ is preferable in comparison with $y$ for testing models\\cite{Busti:2015xqa}.\n\\begin{table}[htbp]\n\\caption{Coverage test for $\\bm{\\theta_1}$ and $\\bm{\\theta_2}$. Refer to the bulk of the text for further details.}\n\\label{tables1}\n\\begin{center}\n\\begin{tabular}{@{}cccccccccccccc@{}}\n\\hline\n& & & & $\\bm{\\theta_1}$ & & & & & & $\\bm{\\theta_2}$ & & \\\\ \\hline \n& \\vline & $y$ & \\vline & \\vline & $z$ & \\vline & & \\vline & $y$ & \\vline & \\vline & $z$ & \\vline \\\\\n\\hline & $1\\sigma$ & $2\\sigma$ &\n 3$\\sigma$ & $1\\sigma$ & $2\\sigma$ &\n 3$\\sigma$ & & $1\\sigma$ & $2\\sigma$ &\n 3$\\sigma$ & $1\\sigma$ & $2\\sigma$ &\n 3$\\sigma$\n\\\\ \\hline\n$q_0$ & 26 & 32 & 42 & 67 & 27 & 6 & & 82 & 12 & 6 & 82 & 18 & 0 \\\\ \n$j_0$ & 10 & 45 & 45 & 64 & 29 & 7 & & 93 & 5 & 2 & 88 & 12 & 0 \\\\ \n$s_0$ & 10 & 67 & 23 & 83 & 15 & 2 & & 92 & 7 & 1 & 93 & 6 & 1 \\\\\n$l_0$ & - & - & - & - & - & - & & 100 & 0 & 0 & 100 & 0 & 0 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\subsection{Testing $\\Lambda$CDM with cosmography}\nSince flat $\\Lambda$CDM model gives unequivocally $j_0=1$ regardless of the matter -and dark energy- content, cosmography can be used as a test for the $\\Lambda$CDM model. In order to show the usefulness of the approach, we have generated Sne Ia data by assuming a XCDM model, different from $\\Lambda$CDM, given by: $H^2\/H_0^2=\\Omega_m (1+z)^3+(1-\\Omega_m)(1+z)^{3(1+w)}$ with $\\Omega_m=0.3$ and $w=-1.3$, which gives $j_0=1.945$. Then, by assuming the variable $z$ and fitting the two parameters sets $\\bm{\\theta_1}$ (fourth order) and $\\bm{\\theta_2}$ (fifth order), we have tested whether the cosmographic approach can rule out $\\Lambda$CDM by obtaining the posterior probability for $j_0$. As depicted in Fig.~\\ref{fig2}, there is some evidence of $j_0 \\neq 1$ when considering $\\bm{\\theta_1}$ but such evidence disappears when $\\bm{\\theta_2}$ is assumed. In addition, we have also included the posterior probability for $j_0$ while considering directly the expression of the Hubble parameter for the XCDM model, which leads to a much better fit. Consequently, the constraints obtained by using cosmography present clear limits while comparing the concordance model with close-enough competitors. \n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{j0_posterior_3cases.eps}\n\\end{center}\n\\caption{Posterior probability for $j_0$ considering 4 parameters $(\\bm{\\theta_1})$, 5 parameters $(\\bm{\\theta_1})$ and XCDM model\\cite{Busti:2015xqa}.}\n\\label{fig2}\n\\end{figure*}\n\\subsection{Reconstructing dark energy models}\nCosmography have been also used for reconstructing some particular models for dark energy, since the underlying action can be expanded around $z=0$ and a correspondence with the cosmographic parameters is obtained. For instance, by considering the usual quintessence\/phantom scalar field model for dark energy, $\\mathcal{S}=\\int {\\rm d}^4x^{}\\sqrt{-g}\\left[-\\frac{1}{2} \\omega (\\phi)\\partial_{\\mu} \\phi \\partial^{\\mu }\\phi -V(\\phi )\\right]\\ ,$ where $\\omega(\\phi)$ is the factor that renormalises the scalar field $\\phi$ and $V(\\phi)$ \nits potential. Then, the derivatives of the potential evaluated today, i.e., at redshift zero, can be expressed in terms of the cosmographic parameters as follows\n\\begin{eqnarray}\n\\label{DE3}\n&\\frac{V_0}{H_0^2}&=2-q_0-\\frac{3\\Omega_m}{2}\\ , \\quad \\frac{V_{z0}}{H_0^2}=4+3q_0-j_0-\\frac{9\\Omega_m}{2}\\ , \\nonumber \\\\\n&\\frac{V_{2z0}}{H_0^2}&=4+8q_0+j_0(4+q_0)+s_0-9\\Omega_m\\ ,\\nonumber \\\\\n\\end{eqnarray}\nwhere we have used the FLRW equations for this model. We can assume $\\Omega_m \\approx 2\/3(1+q_0)$ by considering the universe to be close enough to $\\Lambda$CDM today, what yields a one-to-one correspondence between the derivatives of the potential and the cosmographic parameters. Then, fitting the cosmographic parameters leads to constraints over the derivatives of the scalar potential. However, as shown by Dunsby et al (2015)\\cite{Busti:2015xqa}, the constraints obtained by means of the cosmographic approach provides larger errors than other model-independent methods.\\\\\nSimilarly, higher-order derivatives models as Galileons or $f(R)$ gravities can be evaluated at $z=0$ in terms of the cosmographic parameters. However, in this case the higher number of degrees of freedom does not allow to get a one-to-one correspondence as in (\\ref{DE3}). In order to show this, let us consider $f(R)$ gravity, whose derivatives evaluated today lead to\n\\begin{eqnarray}\n\\label{DE4}\n&&\\frac{f_0}{6H_0^2}\\,=\\,-\\alpha q_0 +\\Omega_m + 6\\beta\\left( 2+q_0-j_0\\right)\\ , \\quad \\frac{f_{z0}}{6H_0^2}\\,=\\,\\alpha\\left(2+q_0-j_0\\right)\\ , \\label{f2z0} \\\\\n&&\\frac{f_{2z0}}{6H_0^2}\\,=\\,6\\beta\\left(2+q_0-j_0\\right)^2+\\alpha\\left[2+4q_0+(2+q_0)j_0+s_0\\right]\\,.\\nonumber \n\\end{eqnarray}\nIn this case, there are two extra free parameters, $\\frac{{\\rm d}f}{{\\rm d}R}|_{R=R_0}=\\alpha$ and $\\frac{{\\rm d}^2f}{{\\rm d}R^2}|_{R=R_0}=\\frac{\\beta}{H_0^2}$.\nThis means we need either theoretical priors or additional tests since data does not provide any constraints over $\\alpha$ and $\\beta$. Some previous works assumed the values of $\\alpha=1$ and $\\beta=0$ a priori, such that the model coincides with General Relativity at $z=0$, but this may lead to instabilities. Let us illustrate the difficulties in getting good constraints for $f(R)$ gravities by generating mock data and assuming some sensible priors over the aforementioned parameters for the following toy-model: $f(R)=R+aR^2+bR^3$ where $\\alpha=2.81$ and $\\beta=0.06$. In Fig.~\\ref{fig3}, the probability for $\\{f_0, f_{z0}, f_{zz0}\\}$ are depicted. Here three different hypotheses have been assumed: the true values of $\\{\\alpha, \\beta\\}$, $\\{\\alpha=1, \\beta=0\\}$ and a ``broad'' marginalisation ($\\alpha \\sim N(1,0.05)$ and $\\beta \\sim N(0.07,0.05)$). The probability of $f_0$ is highly dependent on the choice of $\\{\\alpha, \\beta\\}$ which may even lead to ruling out the true values of $f_0$, whether are not known in advance - as is the case when dealing with real data. There are not large differences for the values of $f_{z0}$ and $f_{zz0}$, but the errors are so large than almost any $f(R)$ may be valid, leading to a completely degenerated result. Consequently cosmography is extremely weak when reconstructing $f(R)$ gravities, since it does not allow to distinguish among different Lagrangians.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{f0_posterior.eps}\n\\includegraphics[width=0.3\\textwidth]{fp0_posterior.eps}\n\\includegraphics[width=0.3\\textwidth]{fpp0_posterior.eps}\n\\end{center}\n\\caption{Probabilities for $f(R)$ and its derivatives evaluated today and the effects of the different choices of the free parameters $\\alpha$ and $\\beta$\\cite{Busti:2015xqa}.}\n\\label{fig3}\n\\end{figure*}\n\\section*{Acknowledgments}\nD.S.-G. acknowledges support from a postdoctoral fellowship Ref.~SFRH\/BPD\/95939\/2013 by Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia (FCT, Portugal) and the support through the research grant UID\/FIS\/04434\/2013 (FCT, Portugal).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}