diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzelyg" "b/data_all_eng_slimpj/shuffled/split2/finalzzelyg" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzelyg" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=7cm]{MilieuPerio.png}\n\\qquad\\begin{tikzpicture}[scale=0.7]\n\\pgfmathsetmacro{\\cubexa}{8}\n\\pgfmathsetmacro{\\cubeya}{1}\n\\pgfmathsetmacro{\\cubeza}{1}\n\n\\draw[black,dotted] (-0.5,-\\cubexa\/2,-0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (-\\cubexa\/2,-0.5,-0.5) -- ++(-0.8,0,0);\n\n\\draw[black,fill=gray!30] (0.5,0.5,-0.5) -- ++(0,0,-\\cubexa\/2+0.5) -- ++(-\\cubeya,0,0) -- ++(0,0,\\cubexa\/2-0.5) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,-0.5) -- ++(0,-\\cubeza,0) -- ++(-\\cubeya,0,0) -- ++(0,\\cubeza,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,-0.5) -- ++(0,0,-\\cubexa\/2+0.5) -- ++(0,-\\cubeza,0) -- ++(0,0,\\cubexa\/2-0.5) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(0,-\\cubexa,0) -- ++(0,0,-\\cubeya) -- ++(0,\\cubexa,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(-\\cubeza,0,0) -- ++(0,0,-\\cubeya) -- ++(\\cubeza,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(0,-\\cubexa,0) -- ++(-\\cubeza,0,0) -- ++(0,\\cubexa,0) -- cycle;\n\n\\draw[black,fill=gray!30] (\\cubexa\/2,0.5,0.5) -- ++(-\\cubexa,0,0) -- ++(0,-\\cubeya,0) -- ++(\\cubexa,0,0) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2,0.5,0.5) -- ++(0,0,-\\cubeza) -- ++(0,-\\cubeya,0) -- ++(0,0,\\cubeza) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2,0.5,0.5) -- ++(-\\cubexa,0,0) -- ++(0,0,-\\cubeza) -- ++(\\cubexa,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(0,-\\cubexa\/2+0.5,0) -- ++(0,0,-\\cubeya) -- ++(0,\\cubexa\/2-0.5,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(-\\cubeza,0,0) -- ++(0,0,-\\cubeya) -- ++(\\cubeza,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2,0.5) -- ++(0,-\\cubexa,0) -- ++(-\\cubeza,0,0) -- ++(0,\\cubexa,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2) -- ++(0,0,-\\cubexa\/2+0.5) -- ++(-\\cubeya,0,0) -- ++(0,0,\\cubexa\/2-0.5) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2) -- ++(0,-\\cubeza,0) -- ++(-\\cubeya,0,0) -- ++(0,\\cubeza,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2) -- ++(0,0,-\\cubexa\/2+0.5) -- ++(0,-\\cubeza,0) -- ++(0,0,\\cubexa\/2-0.5) -- cycle;\n\\draw[black,dotted] (\\cubexa\/2,0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2,-0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2,0.5,-0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2,-0.5,-0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (-\\cubexa\/2,0.5,0.5) -- ++(-0.8,0,0);\n\\draw[black,dotted] (-\\cubexa\/2,-0.5,0.5) -- ++(-0.8,0,0);\n\\draw[black,dotted] (-\\cubexa\/2,0.5,-0.5) -- ++(-0.8,0,0);\n\n\\draw[black,dotted] (0.5,\\cubexa\/2,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,\\cubexa\/2,-0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (-0.5,\\cubexa\/2,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (-0.5,\\cubexa\/2,-0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,-\\cubexa\/2,0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (0.5,-\\cubexa\/2,-0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (-0.5,-\\cubexa\/2,0.5) -- ++(0,-0.8,0);\n\n\\draw[black,dotted] (0.5,0.5,\\cubexa\/2) -- ++(0,0,0.8);\n\\draw[black,dotted] (-0.5,0.5,\\cubexa\/2) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,-0.5,\\cubexa\/2) -- ++(0,0,0.8);\n\\draw[black,dotted] (-0.5,-0.5,\\cubexa\/2) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,0.5,-\\cubexa\/2) -- ++(0,0,-0.8);\n\\draw[black,dotted] (-0.5,0.5,-\\cubexa\/2) -- ++(0,0,-0.8);\n\\draw[black,dotted] (0.5,-0.5,-\\cubexa\/2) -- ++(0,0,-0.8);\n\\draw[black,dotted] (-0.5,-0.5,-\\cubexa\/2) -- ++(0,0,-0.8);\n\\end{tikzpicture}\n\\caption{Left: unbounded periodic cubic lattice made of thin bars of width $\\varepsilon$ with square cross-sections. Right: geometry $\\Omega$ of the near field, aka boundary layer, problem at the junction regions. \\label{PeriodicLattice}}\n\\end{figure}\n\nWith the profusion of works related to graphene in physics, an important effort has been made in the mathematical community to understand the asymptotic behaviour of the spectrum of the Dirichlet operator in quantum waveguides made of thin ligaments, of characteristic width $\\varepsilon\\ll1$, forming unbounded periodic lattices. Various geometries have been considered and we refer the reader to \\cite{Kuch02} for a review article. \nTo address such problems, the general approach can be summarized as follows. Using the Floquet-Bloch-Gelfand theory \\cite{Gelf50,Kuch82,Skri87,Kuch93}, one shows that the spectrum of the operator in the periodic domain has a band-gap structure, the bands being generated by the eigenvalues of a spectral problem set on the periodicity cell with quasi-periodic boundary conditions involving the Floquet-Bloch parameter. The first step consists in applying techniques of dimension reduction to derive a 1D model for this spectral problem on the periodicity cell. Then one studies precisely this 1D model depending on the Floquet-Bloch parameter to get information on the spectral bands.\\\\\n\\newline\nLet us mention that in the past, slapdash and casual conclusions have been made concerning the 1D model problem. This model consists of ordinary differential equations on the ligaments obtained when taking $\\varepsilon\\to0$ supplemented by transmission conditions at the nodes of the graph. Certain authors have inappropriately applied L. Pauling's model \\cite{Paul36} and imposed Kirchoff transmission conditions at the nodes. These Kirchoff conditions boil down to impose continuity of the field and zero outgoing flux (the sum of the derivatives of the field along the outgoing directions at the node vanishes). This is correct for the Laplacian with Neumann boundary conditions (BC) and has been rigorously justified in \\cite{KuZe03,ExPo05,Post12}. However it has been shown by D. Grieser in \\cite{Grie08} (see also \\cite{MoVa07}) that for the Dirichlet problem, in general the right conditions to impose at the nodes are Dirichlet ones. More precisely, it has been proved in \\cite{Grie08} that the transmission conditions to impose depend on the existence or absence of so-called threshold resonances for the near field operator defined as the Laplacian in the geometry obtained when zooming at the junction regions (denoted $\\Omega$ in the sequel, see Figure \\ref{PeriodicLattice} right). We say that there is a threshold resonance if there is a non zero bounded function which solves the homogeneous problem at the frequency coinciding with the bottom of the essential spectrum (the threshold) of the Laplace operator. For the Neumann problem, the threshold is $\\Lambda_\\dagger=0$ and there is a threshold resonance because the constants solve $-\\Delta u=0$ in $\\Omega$ + Neumann BC. Due to this property, one must impose Kirchoff conditions at the nodes. For the Dirichlet problem, the continuous spectrum starts at a positive threshold $\\Lambda_\\dagger>0$ and in general the only solution to the problem \n\\begin{equation}\\label{PbThreshold}\n\\begin{array}{|rcll}\n-\\Delta u&=&\\Lambda_\\dagger u &\\mbox{ in } \\Omega\\\\[3pt]\nu&=&0&\\mbox{ on } \\partial\\Omega\n\\end{array}\n\\end{equation}\nwhich remains bounded at infinity is zero, i.e. there is no threshold resonance. Because of this feature, one should impose Dirichlet condition at the nodes of the 1D model (see (\\ref{farfield_BC}) for the precise moment where this pops up in the analysis below).\n\n\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{tikzpicture}[scale=1]\n\\draw[fill=gray!30,draw=none](-2,-1\/2) rectangle (2,1\/2);\n\\draw[fill=gray!30,draw=none](-1\/2,-1\/2) rectangle (1\/2,2);\n\\draw[black] (-2,-1\/2)--(2,-1\/2);\n\\draw[black] (-2,1\/2)--(-1\/2,1\/2)--(-1\/2,2);\n\\draw[black] (2,1\/2)--(1\/2,1\/2)--(1\/2,2);\n\\draw[black,dotted] (-2.5,-1\/2)--(2.5,-1\/2);\n\\draw[black,dotted] (-2.5,1\/2)--(-2,1\/2);\n\\draw[black,dotted] (2.5,1\/2)--(2,1\/2);\n\\draw[black,dotted] (-1\/2,2.5)--(-1\/2,2);\n\\draw[black,dotted] (1\/2,2.5)--(1\/2,2);\n\\end{tikzpicture}\\qquad\\begin{tikzpicture}[scale=1]\n\\begin{scope}[rotate=20]\n\\draw[fill=gray!30,draw=none](-1\/2,0) rectangle (1\/2,2);\n\\draw[black] (-1\/2,1\/2)--(-1\/2,2);\n\\draw[black] (1\/2,0)--(1\/2,2);\n\\draw[black,dotted] (-1\/2,2.5)--(-1\/2,2);\n\\draw[black,dotted] (1\/2,2.5)--(1\/2,2);\n\\end{scope}\n\\draw[fill=gray!30,draw=none](-2,-1\/2) rectangle (2,1\/2);\n\\begin{scope}[rotate=20]\n\\draw[black,dashed] (0,0)--(0,2.5);\n\\end{scope}\n\\draw[black,dashed] (-2.5,0)--(2.5,0);\n\\draw[black] (-2,-1\/2)--(2,-1\/2);\n\\draw[black] (-2,1\/2)--(-0.7,1\/2);\n\\draw[black] (2,1\/2)--(0.35,1\/2);\n\\draw[black,dotted] (-2.5,-1\/2)--(2.5,-1\/2);\n\\draw[black,dotted] (-2.5,1\/2)--(-2,1\/2);\n\\draw[black,dotted] (2.5,1\/2)--(2,1\/2);\n\\pgfmathparse{.5*cos(110)}\\let\\x\\pgfmathresult\n\\pgfmathparse{.5*sin(110)}\\let\\y\\pgfmathresult\n\\draw[blue,->] (\\x,\\y) arc (110:180:.5cm);\n\\node[left,blue] at (145:.5) {\\small $\\alpha$};\n\\end{tikzpicture}\\qquad\\raisebox{-0.6cm}{\\begin{tikzpicture}[scale=0.75]\n\\draw[fill=gray!30,draw=none](-2,-1\/2) rectangle (2,1\/2);\n\\draw[fill=gray!30,draw=none](-1\/2,-2) rectangle (1\/2,2);\n\\draw[black] (-2,-1\/2)--(-1\/2,-1\/2)--(-1\/2,-2);\n\\draw[black] (2,-1\/2)--(1\/2,-1\/2)--(1\/2,-2);\n\\draw[black] (-2,1\/2)--(-1\/2,1\/2)--(-1\/2,2);\n\\draw[black] (2,1\/2)--(1\/2,1\/2)--(1\/2,2);\n\\draw[black,dotted] (2.5,-1\/2)--(2,-1\/2);\n\\draw[black,dotted] (-2.5,-1\/2)--(-2,-1\/2);\n\\draw[black,dotted] (-2.5,1\/2)--(-2,1\/2);\n\\draw[black,dotted] (2.5,1\/2)--(2,1\/2);\n\\draw[black,dotted] (-1\/2,2.5)--(-1\/2,2);\n\\draw[black,dotted] (1\/2,2.5)--(1\/2,2);\n\\draw[black,dotted] (-1\/2,-2.5)--(-1\/2,-2);\n\\draw[black,dotted] (1\/2,-2.5)--(1\/2,-2);\n\\begin{scope}[rotate=45]\n\\draw[fill=gray!60,draw=none](-0.707,-0.707) rectangle (0.707,0.707);\n\\end{scope}\n\\end{tikzpicture}}\n\\caption{Examples of T- and X-shaped 2D geometries.\\label{2Dexamples}} \n\\end{figure}\n\n\\noindent From there, authors have worked to establish rigorous results showing the absence of non zero bounded solution to (\\ref{PbThreshold}). First, different planar\nquantum waveguides made of T-, X- and Y-shaped junctions of thin ligaments have been considered in \\cite{Naza10a,Naza14,Naza14a,NaRU14,NaRU15,Naza17,Pank17}. In these articles, additionally it has been proved that the near field operator in $\\Omega$, depending on the considered geometry, may have discrete spectrum (one or several eigenvalue below the continuous spectrum). When the latter exists, the low-frequency range of the spectrum in the periodic domain is not described by the above mentioned 1D model with Dirichlet conditions at the nodes. Instead, the first spectral bands, their number being equal to the multiplicity of the discrete spectrum, are generated by functions which are localized at the junctions regions. Let us mention that for certain exceptional geometries, for example for a sequence of angles $\\alpha$ in the central domain of Figure \\ref{2Dexamples}, one may have non zero solutions to (\\ref{PbThreshold}) which remain bounded at infinity, i.e. existence of threshold resonance. In these situations, at least if the dimension of the space of bounded solutions to (\\ref{PbThreshold}) is one, the good 1D model describing the spectral bands which are not associated with the discrete spectrum in $\\Omega$, has certain Kirchoff transmission conditions at the nodes which depend on the geometry. We emphasize that this leads to very different spectra for the operator in the periodic medium. More precisely, when there is no threshold resonance, the bands of the spectrum in the periodic material become very small as $\\varepsilon\\to0$ and the spectral gaps enlarge. In other words, the spectrum becomes rather sparse, for most of the spectral parameters waves cannot propagate and the limit 1D ligaments are somehow disconnected. This is what we will obtain below in our configuration. On the other hand, when Problem (\\ref{PbThreshold}) admits a space of dimension one of bounded solutions, the spectral bands in the periodic material are much larger.\\\\\n\\newline\nAfterwards, 3D geometries were considered in \\cite{BaMaNa17} (see also the corresponding note \\cite{BaMaNa15}). In these articles, the authors consider quantum waveguides for which the near field domain $\\Omega$ is a cruciform junction of two cylinders whose cross section coincides with the unit disc. The passage from the planar case to the spatial case requires the non obvious adaptation of the methods. In particular the characterization of the discrete spectrum of the near field operator and the proof of absence of threshold resonances are much more involved. \nIn the present work, we study an even more intricate 3D geometry for which the near field geometry is the union of three waveguides. Rather precise Friedrichs estimates are required to prove that the discrete spectrum of the near field operator contains exactly one eigenvalue and to show that at the threshold, zero is the only bounded solution. This work also complements the study of \\cite{BaMaNa17} thanks to the numerical experiments. Let us mention that the cruciform junction of two 3D cylinders with square cross section reduces to a 2D problem in a X-shaped geometry (see again Figure \\ref{2Dexamples} right) and due to factoring out, is of no interest.\\\\\n\\newline\nThe outline is as follows. First we describe the problem, introduce the notation and present the main results. Then we study the discrete spectrum of the near field operator. In section \\ref{sectionAbsenceofBoundedSol}, we demonstrate the absence of threshold resonance for the near field operator. Section \\ref{SectionModels} is dedicated to the analysis of the main theorem of the article (Theorem \\ref{MainThmPerio}) with the derivation of asymptotic models for the spectral bands in the original periodic domain. Finally we show some numerics to complement the results and conclude with some appendix containing the proof of two lemmas needed in sections \\ref{Section_DiscreteSpectrum}, \\ref{sectionAbsenceofBoundedSol}. \n\n\\section{Notation and main results}\n\nFor $j=1,2,3$, introduce the cylinder with square cross section\n\\begin{equation}\\label{defBars}\nL_j := \\{x:=(x_1,x_2,x_2)\\in\\mathbb{R}^3\\,|\\,|x_k|<1\/2\\mbox{ for }k\\in\\{1,2,3\\}\\setminus\\{j\\}\\}\n\\end{equation}\nand for $\\varepsilon>0$ small, $m,n\\in\\mathbb{Z}:=\\{0,\\pm1,\\pm2,\\dots\\}$, set\n\\[\n\\begin{array}{rcl}\nL^{mn\\varepsilon}_1 & := & \\{x\\in\\mathbb{R}^3\\,|\\,|x_2-m|<\\varepsilon\/2,\\,|x_3-n|<\\varepsilon\/2\\}\\\\[2pt]\nL^{mn\\varepsilon}_2 & := & \\{x\\in\\mathbb{R}^3\\,|\\,|x_1-m|<\\varepsilon\/2,\\,|x_3-n|<\\varepsilon\/2\\}\\\\[2pt]\nL^{mn\\varepsilon}_3 & := & \\{x\\in\\mathbb{R}^3\\,|\\,|x_1-m|<\\varepsilon\/2,\\,|x_2-n|<\\varepsilon\/2\\}.\n\\end{array}\n\\]\nFinally define the unbounded periodic domain \n\\[\n\\Pi^\\varepsilon:=\\bigcup_{m,n\\in\\mathbb{Z}}L^{mn\\varepsilon}_1\\cup L^{mn\\varepsilon}_2\\cup L^{mn\\varepsilon}_3\n\\]\n(see Figure \\ref{PeriodicLattice} left). Consider the Dirichlet spectral problem for the Laplace operator \n\\begin{equation}\\label{PbSpectral}\n\\begin{array}{|rcll}\n-\\Delta u^\\varepsilon&=&\\lambda^\\varepsilon\\,u^\\varepsilon &\\mbox{ in }\\Pi^\\varepsilon\\\\[3pt]\nu^\\varepsilon&=&0&\\mbox{ on } \\partial\\Pi^\\varepsilon.\n\\end{array}\n\\end{equation}\nThe variational form associated with this problem writes\n\\begin{equation}\\label{variationPb}\n\\int_{\\Pi^\\varepsilon}\\nabla u^\\varepsilon\\cdot\\nabla v^\\varepsilon\\,dx=\\lambda^\\varepsilon\\int_{\\Pi^\\varepsilon} u^\\varepsilon v^\\varepsilon\\,dx,\\qquad\\forall v^\\varepsilon\\in\\mrm{H}^1_0(\\Pi^\\varepsilon).\n\\end{equation}\nHere $\\mrm{H}^1_0(\\Pi^\\varepsilon)$ stands for the Sobolev space of functions of $\\mrm{H}^1(\\Pi^\\varepsilon)$ which vanish on the boundary $\\partial\\Pi^\\varepsilon$. Classically (see e.g \\cite[\\S10.1]{BiSo87}), the variational problem (\\ref{variationPb}) gives rise to an unbounded, positive definite, selfadjoint operator $A^\\varepsilon$ in the Hilbert space $\\mrm{L}^2(\\Pi^\\varepsilon)$, with domain $\\mathcal{D}(A^\\varepsilon)\\subset\\mrm{H}^1_0(\\Pi^\\varepsilon)$. Note that this operator is sometimes called the quantum graph Laplacian \\cite{BeKu13,Post12}. Since $\\Pi^\\varepsilon$ is unbounded, the embedding $\\mrm{H}^1_0(\\Pi^\\varepsilon)\\subset \\mrm{L}^2(\\Pi^\\varepsilon)$ is not compact and $A^\\varepsilon$ has a non empty essential component $\\sigma_e(A^\\varepsilon)$ (\\cite[Thm. 10.1.5]{BiSo87}). Actually, due to the periodicity, we have $\\sigma_e(A^\\varepsilon)=\\sigma(A^\\varepsilon)$. The Gelfand transform (see the surveys \\cite{Kuch82,Naza99a} and books \\cite{Skri87,Kuch93}) \n\\[\nu^{\\varepsilon}(x)\\mapsto U^\\varepsilon(x,\\eta)=\\cfrac{1}{(2\\pi)^{3\/2}}\\sum_{\\iota\\in\\mathbb{Z}^3} e^{i\\eta\\cdot \\iota}u^{\\varepsilon}(x+\\iota),\\quad\\eta=(\\eta_1,\\eta_2,\\eta_3),\n\\]\nchanges Problem (\\ref{PbSpectral}) into the following spectral problem with quasi-periodic at the faces located at $x_1=\\pm1\/2$, $x_2=\\pm1\/2$, $x_3=\\pm1\/2$,\n\\begin{equation}\\label{PbSpectralCell}\n\\begin{array}{|rcll}\n-\\Delta U^\\varepsilon(x,\\eta)&=&\\Lambda^\\varepsilon\\,U^\\varepsilon(x,\\eta) & x\\in\\omega^\\varepsilon\\\\[3pt]\nU^\\varepsilon(x,\\eta)&=&0&x\\in\\partial\\omega^\\varepsilon\\cap\\partial\\Pi^\\varepsilon\\\\[3pt]\nU^\\varepsilon(-1\/2,x_2,x_3,\\eta) & = & e^{i\\eta_1}U^\\varepsilon(+1\/2,x_2,x_3,\\eta) & (x_2,x_3)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\n\\partial_{x_1}U^\\varepsilon(-1\/2,x_2,x_3,\\eta) & = & e^{i\\eta_1}\\partial_{x_1}U^\\varepsilon(+1\/2,x_2,x_3,\\eta) & (x_2,x_3)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\nU^\\varepsilon(x_1,-1\/2,x_3,\\eta) & = & e^{i\\eta_2}U^\\varepsilon(x_1,+1\/2,x_3,\\eta) & (x_1,x_3)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\n\\partial_{x_2}U^\\varepsilon(x_1,-1\/2,x_3,\\eta) & = & e^{i\\eta_2}\\partial_{x_2}U^\\varepsilon(x_1,+1\/2,x_3,\\eta) & (x_1,x_3)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\nU^\\varepsilon(x_1,x_2,-1\/2,\\eta) & = & e^{i\\eta_3}U^\\varepsilon(x_1,x_2,+1\/2,\\eta) & (x_1,x_2)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[3pt]\n\\partial_{x_3}U^\\varepsilon(x_1,x_2,-1\/2,\\eta) & = & e^{i\\eta_3}\\partial_{x_3}U^\\varepsilon(x_1,x_2,+1\/2,\\eta) & (x_1,x_2)\\in(-\\varepsilon\/2;\\varepsilon\/2)^2,\n\\end{array}\n\\end{equation}\nset in the periodicity cell \n\\[\n\\omega^\\varepsilon:=\\omega^\\varepsilon_1\\cup\\omega^\\varepsilon_2\\cup\\omega^\\varepsilon_3\\quad\\mbox{ with }\\quad\\begin{array}{|l}\n\\omega^\\varepsilon_1:=(-1\/2;1\/2)\\times(-\\varepsilon\/2;\\varepsilon\/2)^2\\\\[2pt]\n\\omega^\\varepsilon_2:=(-\\varepsilon\/2;\\varepsilon\/2)\\times(-1\/2;1\/2)\\times(-\\varepsilon\/2;\\varepsilon\/2)\\\\[2pt]\n\\omega^\\varepsilon_3:=(-\\varepsilon\/2;\\varepsilon\/2)^2\\times(-1\/2;1\/2).\n\\end{array}\n\\]\nProblem (\\ref{PbSpectralCell}) is formally selfadjoint for any value of the dual variable $\\eta\\in\\mathbb{R}^3$. Additionally it is $2\\pi$-periodic with respect to each of the $\\eta_j$ because the transformation $\\eta_j\\mapsto\\eta_j+2\\pi$ leaves invariant the quasiperiodicity conditions. For any $\\eta\\in[0;2\\pi)^3$, the spectrum of (\\ref{PbSpectralCell}) is discrete, made of a monotone increasing positive sequence of eigenvalues\n\\[\n0<\\Lambda_1^\\varepsilon(\\eta)\\le \\Lambda_2^\\varepsilon(\\eta)\\le \\dots \\le \\Lambda_p^\\varepsilon(\\eta)\\le \\dots\n\\]\nwhere the $\\Lambda_p^\\varepsilon(\\eta)$ are counted according to their multiplicity. The functions \n\\[\n\\eta\\mapsto \\Lambda_p^\\varepsilon(\\eta)\n\\]\nare continuous (\\cite[Chap. 9]{Kato95}) so that the sets \n\\begin{equation}\\label{SpectralBands}\n\\Upsilon^\\varepsilon_p=\\{\\Lambda_p^\\varepsilon(\\eta),\\,\\eta\\in[0;2\\pi)^3\\}\n\\end{equation}\nare connected compact segments. Finally, according to the theory (see again \\cite{Gelf50,Kuch82,Skri87,Naza99a,Kuch93}), the spectrum of the operator $A^{\\varepsilon}$ has the form\n\\[\n\\sigma(A^\\varepsilon)=\\bigcup_{p\\in\\mathbb{N}^\\ast}\\Upsilon^\\varepsilon_p\n\\]\nwhere $\\mathbb{N}^\\ast:=\\{1,2,\\dots\\}$. At this stage, we see that to clarify the behaviour of the spectrum of $A^\\varepsilon$ with respect to $\\varepsilon\\to0^+$, we need to study the dependence of the $\\Upsilon^\\varepsilon_p$ with respect to $\\varepsilon$.\\\\\n\\newline\nAs already mentioned in the introduction, the analysis developed for example in \\cite{Grie08,Naza05,Naza14} shows that the asymptotic behaviour of the $\\Upsilon^\\varepsilon_p$ with respect to $\\varepsilon$ depends on the features of the Dirichlet Laplacian in the geometry obtained when zooming at the junction region of the periodicity cell $\\omega^\\varepsilon$. More precisely, introduce the unbounded domain\n\\begin{equation}\\label{DefOmega}\n\\Omega:=L_1\\cup L_2\\cup L_3\n\\end{equation}\n(see Figure \\ref{PeriodicLattice} right) where the $L_j$ are the cylinders with unit square cross section appearing in (\\ref{defBars}). In $\\Omega$, consider the Dirichlet spectral problem for the Laplace operator \n\\begin{equation}\\label{PbSpectralZoom}\n\\begin{array}{|rcll}\n-\\Delta v&=&\\mu\\,v &\\mbox{ in }\\Omega\\\\[3pt]\nv&=&0&\\mbox{ on } \\partial\\Omega\n\\end{array}\n\\end{equation}\nwhich is now independent of $\\varepsilon$. We denote by $A^\\Omega$ the unbounded, positive definite, selfadjoint operator naturally associated with this problem defined in $\\mrm{L}^2(\\Omega)$ and of domain $\\mathcal{D}(A^\\Omega)\\subset\\mrm{H}^1_0(\\Omega)$. Its continuous spectrum $\\sigma_c(A^\\Omega)$ occupies the ray $[2\\pi^2;+\\infty)$ and the threshold point $\\Lambda_\\dagger:=2\\pi^2$ is the first eigenvalue of the Dirichlet problem in the cross sections of the branches of $\\Omega$ (which are unit squares). The main goal of this article is to show the following results.\n\\begin{theorem}\\label{ThmDiscreteSpectrum}\nThe discrete spectrum of the operator $A^{\\Omega}$ contains exactly one eigenvalue $\\mu_1\\in(0;2\\pi^2)$.\n\\end{theorem}\n\\begin{remark}\nNumerically, in Section \\ref{SectionNumerics}, we find $\\mu_1\\approx 12.9\\approx1.3\\pi^2$. Note that the eigenfunctions associated with $\\mu_1$ decay at infinity as $O(-\\sqrt{2\\pi^2-\\mu_1}\\,|x_j|)$ in $L_j$, $j=1,2,3$.\n\\end{remark}\n\\noindent As classical in literature, we shall say that there is a threshold resonance for Problem (\\ref{PbSpectralZoom}) if there is non trivial function which solves (\\ref{PbSpectralZoom}) with the threshold value $\\mu=2\\pi^2$ of the spectral parameter.\n\\begin{theorem}\\label{ThmNoBoundedSol}\nThere is no threshold resonance for Problem (\\ref{PbSpectralZoom}). \n\\end{theorem}\n\\noindent From Theorems \\ref{ThmDiscreteSpectrum} and \\ref{ThmNoBoundedSol}, we will derive the final result for the spectrum of $A^\\varepsilon$ in the initial unbounded periodic lattice:\n\\begin{theorem}\\label{MainThmPerio}\nThere are positive values $\\varepsilon_p>0$ and $c_p>0$ such that for the spectral bands introduced in (\\ref{SpectralBands}), we have the estimates\n\\[\n\\begin{array}{ll}\n\\Upsilon^\\varepsilon_1\\subset \\varepsilon^{-2}\\mu_1+\\varepsilon^{-2}e^{-\\sqrt{2\\pi^2-\\mu_1}\/\\varepsilon}(-c_1;c_1),&\\ \\varepsilon\\in(0;\\varepsilon_1]\\\\[8pt]\n\\Upsilon^\\varepsilon_{1+q+3(p-1)}\\subset \\varepsilon^{-2}2\\pi^2+p^2\\pi^2+\\varepsilon\\,(-c_p;c_p),&\\ \\varepsilon\\in(0;\\varepsilon_p],\\ q=1,2,3,\\ p\\in\\mathbb{N}^\\ast.\\\\\n\\end{array}\n\\]\n\\end{theorem}\n\\noindent Let us comment this result. First, as already mentioned in the introduction, the spectrum of the operator $A^\\varepsilon$ goes to $+\\infty$ as $\\varepsilon\\to0$. Additionally this spectrum becomes very sparse. Indeed Theorem \\ref{MainThmPerio} implies the following results. The length of the spectral bands are infinitesimal as $\\varepsilon\\to0$. Moreover, between the bands $\\Upsilon^\\varepsilon_1$ and $\\Upsilon^\\varepsilon_2$, there is a gap, that is a segment of spectral parameters $\\Lambda$ such that waves cannot propagate, of size $O(\\varepsilon^{-2}(2\\pi^2-\\mu_1))$ while between $\\Upsilon^\\varepsilon_{1+q+3(p-1)}$ and $\\Upsilon^\\varepsilon_{1+q+3p}$, the gap is of width $O(\\pi^2(2p-1))$. Therefore the main message here is that in the thin lattice $\\Pi^\\varepsilon$ the propagation of waves is hampered and occurs for very narrow intervals of frequencies.\n\n\n\\section{Properties of the discrete spectrum of $A^{\\Omega}$}\\label{Section_DiscreteSpectrum}\nThe goal of this section is to prove Theorem \\ref{ThmDiscreteSpectrum}. We start by showing that the discrete spectrum of $A^{\\Omega}$ is non empty (for related multidimensional problems, see \\cite{Naza12a}). \n\\begin{proposition}\\label{PropoAtLeastOne}\nThe discrete spectrum of $A^{\\Omega}$ has at least one eigenvalue.\n\\end{proposition}\n\\begin{proof}\nDefine the 2D X-shaped geometry $\\Omega^{2D}:=\\Omega^{2D}_1\\cup\\Omega^{2D}_2$ with $\\Omega^{2D}_1:=\\mathbb{R}\\times(-1\/2;1\/2)$ and $\\Omega^{2D}_2:=(-1\/2;1\/2)\\times\\mathbb{R}$ (see Figure \\ref{2Dexamples} right). According to \\cite{ScRW89,ABGM91} (see also \\cite[Thm. 2.1]{Naza14b} as well as the discussion at the end of this proof), we know that the Dirichlet Laplacian in $\\Omega^{2D}$ admits exactly one eigenvalue $\\mu^{2D}$ below the continuous spectrum which coincides with $[\\pi^2;+\\infty)$. Let $\\varphi\\in\\mrm{H}^1_0(\\Omega^{2D})$ be a corresponding eigenfunction. In the domain $\\Omega\\subset\\mathbb{R}^3$ introduced in (\\ref{DefOmega}), consider the function $v$ such that\n\\[\n\\begin{array}{ll}\nv(x_1,x_2,x_3) = \\varphi(x_1,x_2)\\cos(\\pi x_3)\\ \\mbox{ in }L_1\\cup L_2,\\qquad\\qquad &v(x_1,x_2,x_3)=0\\ \\mbox{ in }\\Omega\\setminus\\overline{L_1\\cup L_2}.\n\\end{array}\n\\]\nWe have \n\\[\n\\int_{\\Omega}|\\nabla v|^2\\,dx=\\int_{L_1\\cup L_2}|\\nabla v|^2\\,dx=-\\int_{L_1\\cup L_2}\\Delta v v\\,dx=(\\mu^{2D}+\\pi^2)\\int_{\\Omega} v^2\\,dx.\n\\]\nBut according to the max-min principle (cf. \\cite[Thm. 10.2.2]{BiSo87}), we know that\n\\begin{equation}\\label{CaracInfSup}\n\\inf \\sigma(A^{\\Omega})=\\inf_{w\\in\\mrm{H}^1_0(\\Omega)\\setminus\\{0\\}}\\cfrac{\\displaystyle\\int_{\\Omega}|\\nabla w|^2\\,dx}{\\displaystyle\\int_{\\Omega}w^2\\,dx}\\,.\n\\end{equation}\nInserting the above $v$ in the right hand side of (\\ref{CaracInfSup}), we deduce that the discrete spectrum of $A^{\\Omega}$ contains an eigenvalue below $\\mu^{2D}+\\pi^2$.\\\\\n\\newline\nWe end this proof by describing the elegant trick proposed in \\cite{ScRW89,ABGM91} to show the existence of an eigenvalue in the discrete spectrum of the Dirichlet Laplacian in $\\Omega^{2D}$. Consider the square coloured in dark gray of Figure \\ref{2Dexamples} right. It has side $\\sqrt{2}$. Therefore the first eigenvalue of the Dirichlet Laplacian in this geometry is equal to $\\pi^2$. Hence, in any domain containing strictly this square and included in $\\Omega^{2D}$, the first eigenvalue of the Dirichlet Laplacian is strictly less than $\\pi^2$. Extending a corresponding eigenfunction by zero to $\\Omega^{2D}$ and using the max-min principle, we infer that the discrete spectrum of the Dirichlet Laplacian in $\\Omega^{2D}$ is not empty.\n\\end{proof}\n\n\n\\noindent It remains to show that the discrete spectrum of $A^{\\Omega}$ has at most one eigenvalue. To proceed, first we establish a result of symmetry. \n\\begin{lemma}\\label{LemmaSym}\nLet $v\\in\\mrm{H}^1_0(\\Omega)$ be an eigenfunction of the operator $A^{\\Omega}$ associated with an eigenvalue $\\mu<2\\pi^2$. Then $v$ is symmetric with respect to the three planes $(Ox_2x_3)$, $(Ox_3x_1)$, $(Ox_1x_2)$.\n\\end{lemma}\n\\begin{proof}\nWe present the proof of symmetry with respect to the plane $(Ox_2x_3)$. The two other symmetries can be established similarly. Introduce the function $\\varphi$ such that\n\\[\n\\varphi(x_1,x_2,x_3)=v(x_1,x_2,x_3)-v(-x_1,x_2,x_3).\n\\]\nOur goal is to prove that $\\varphi\\equiv0$. Set $\\Omega^+_1:=\\{(x_1,x_2,x_3)\\in\\Omega\\,|\\,x_1>0\\}$. The function $\\varphi$ satisfies $-\\Delta\\varphi=\\mu\\varphi$ in $\\Omega^+_1$ and $\\varphi=0$ on $\\partial\\Omega^+_1$. Therefore we have \n\\begin{equation}\\label{IdentityEigen}\n\\int_{\\Omega^+_1}|\\nabla\\varphi|^2\\,dx=\\mu\\int_{\\Omega^+_1}\\varphi^2\\,dx.\n\\end{equation}\n\n\n\\begin{figure}[!ht]\n\\centering\\begin{tikzpicture}[scale=0.75]\n\\pgfmathsetmacro{\\cubexa}{8}\n\\pgfmathsetmacro{\\cubeya}{1}\n\\pgfmathsetmacro{\\cubeza}{1}\n\\begin{scope}[rotate=90]\n\\draw[black,dotted] (0,-\\cubexa\/2,-0.5) -- ++(0,-0.8,0);\n\n\\draw[black,fill=gray!30] (0.5,0.5,-1) -- ++(0,0,-\\cubexa\/2) -- ++(-\\cubeya\/2,0,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,-1) -- ++(0,-\\cubeza,0) -- ++(-\\cubeya\/2,0,0) -- ++(0,\\cubeza,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,-1) -- ++(0,0,-\\cubexa\/2) -- ++(0,-\\cubeza,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(0,0,-\\cubeya) -- ++(0,\\cubexa\/2,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(-\\cubeza\/2,0,0) -- ++(0,0,-\\cubeya) -- ++(\\cubeza\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(-\\cubeza\/2,0,0) -- ++(0,\\cubexa\/2,0) -- cycle;\n\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(-\\cubexa\/2,0,0) -- ++(0,-\\cubeya,0) -- ++(\\cubexa\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(0,0,-\\cubeza) -- ++(0,-\\cubeya,0) -- ++(0,0,\\cubeza) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(-\\cubexa\/2,0,0) -- ++(0,0,-\\cubeza) -- ++(\\cubexa\/2,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(-0.5,0,0) -- ++(0,-\\cubeya,0) -- ++(0.5,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(0,0,-\\cubeza) -- ++(0,-\\cubeya,0) -- ++(0,0,\\cubeza) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(-0.5,0,0) -- ++(0,0,-\\cubeza) -- ++(0.5,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,-1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(0,0,-\\cubeya) -- ++(0,\\cubexa\/2,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,-1,0.5) -- ++(-\\cubeza\/2,0,0) -- ++(0,0,-\\cubeya) -- ++(\\cubeza\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,-1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(-\\cubeza\/2,0,0) -- ++(0,\\cubexa\/2,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,-\\cubexa\/2) -- ++(-\\cubeya\/2,0,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,-\\cubeza,0) -- ++(-\\cubeya\/2,0,0) -- ++(0,\\cubeza,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,-\\cubexa\/2) -- ++(0,-\\cubeza,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\n\\draw[black,dotted] (\\cubexa\/2+1,0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,-0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,0.5,-0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,-0.5,-0.5) -- ++(0.8,0,0);\n\n\\draw[black,dotted] (0.5,\\cubexa\/2+1,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,\\cubexa\/2+1,-0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0,\\cubexa\/2+1,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0,\\cubexa\/2+1,-0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,-\\cubexa\/2-1,0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (0.5,-\\cubexa\/2-1,-0.5) -- ++(0,-0.8,0);\n\\draw[black,dotted] (0,-\\cubexa\/2-1,0.5) -- ++(0,-0.8,0);\n\n\\draw[black,dotted] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0,0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,-0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0,-0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,0.5,-\\cubexa\/2-1) -- ++(0,0,-0.8);\n\\draw[black,dotted] (0,0.5,-\\cubexa\/2-1) -- ++(0,0,-0.8);\n\\draw[black,dotted] (0.5,-0.5,-\\cubexa\/2-1) -- ++(0,0,-0.8);\n\n\\node at (3,0,0.5){$L_1^+$};\n\\node at (0,2.6,-1){$S_2^+$};\n\\node at (0,-3.6,-1){$S_2^-$};\n\\node at (0,-0.5,2){$S_3^+$};\n\\node at (0,-0.5,-4.5){$S_3^-$};\n\\draw[black,->] (1.6,-1.6,0) -- ++(-1,1,0);\n\\node at (1.8,-2,0){$Q^+_1$};\n\\end{scope}\n\\end{tikzpicture}\\qquad \\begin{tikzpicture}[scale=0.75]\n\\pgfmathsetmacro{\\cubexa}{8}\n\\pgfmathsetmacro{\\cubeya}{1}\n\\pgfmathsetmacro{\\cubeza}{1}\n\\begin{scope}[rotate=90]\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(0,0,-\\cubeya\/2) -- ++(0,\\cubexa\/2,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(-\\cubeza\/2,0,0) -- ++(0,0,-\\cubeya\/2) -- ++(\\cubeza\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,\\cubexa\/2+1,0.5) -- ++(0,-\\cubexa\/2,0) -- ++(-\\cubeza\/2,0,0) -- ++(0,\\cubexa\/2,0) -- cycle;\n\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(-\\cubexa\/2,0,0) -- ++(0,-\\cubeya\/2,0) -- ++(\\cubexa\/2,0,0) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(0,0,-\\cubeza\/2) -- ++(0,-\\cubeya\/2,0) -- ++(0,0,\\cubeza\/2) -- cycle;\n\\draw[black,fill=gray!30] (\\cubexa\/2+1,0.5,0.5) -- ++(-\\cubexa\/2,0,0) -- ++(0,0,-\\cubeza\/2) -- ++(\\cubexa\/2,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(-0.5,0,0) -- ++(0,-\\cubeya\/2,0) -- ++(0.5,0,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(0,0,-\\cubeza\/2) -- ++(0,-\\cubeya\/2,0) -- ++(0,0,\\cubeza\/2) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,0.5) -- ++(-0.5,0,0) -- ++(0,0,-\\cubeza\/2) -- ++(0.5,0,0) -- cycle;\n\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,-\\cubexa\/2) -- ++(-\\cubeya\/2,0,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,-\\cubeza\/2,0) -- ++(-\\cubeya\/2,0,0) -- ++(0,\\cubeza\/2,0) -- cycle;\n\\draw[black,fill=gray!30] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,-\\cubexa\/2) -- ++(0,-\\cubeza\/2,0) -- ++(0,0,\\cubexa\/2) -- cycle;\n\n\\draw[black,dotted] (\\cubexa\/2+1,0.5,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,0,0.5) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,0.5,0) -- ++(0.8,0,0);\n\\draw[black,dotted] (\\cubexa\/2+1,0,0) -- ++(0.8,0,0);\n\n\\draw[black,dotted] (0.5,\\cubexa\/2+1,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0.5,\\cubexa\/2+1,0) -- ++(0,0.8,0);\n\\draw[black,dotted] (0,\\cubexa\/2+1,0.5) -- ++(0,0.8,0);\n\\draw[black,dotted] (0,\\cubexa\/2+1,0) -- ++(0,0.8,0);\n\n\\draw[black,dotted] (0.5,0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0,0.5,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0.5,0,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,dotted] (0,0,\\cubexa\/2+1) -- ++(0,0,0.8);\n\\draw[black,->] (1.6,1.6,0) -- ++(-1,-1,0);\n\\node at (1.8,1.8,0){$Q^+$};\n\\end{scope}\n\\end{tikzpicture}\n\\caption{Left: exploded decomposition of $\\Omega^+_1$. Right: exploded decomposition of $\\Omega^+$. \\label{ExplodedDomain}}\n\\end{figure}\n\n\\noindent Define the domains \n\\[\n\\begin{array}{rcl}\nL^+_1 & := & \\{x\\in L_1\\,|\\,x_1>1\/2\\}\\\\[2pt]\nS_2 & := & \\{x\\in L_2\\,|\\,x_1>0\\}=\\Omega^+_1\\cap L_2\\\\[2pt]\nS_3 & := & \\{x\\in L_3\\,|\\,x_1>0\\}=\\Omega^+_1\\cap L_3\\\\[2pt]\nS^{\\pm}_2 & := & \\{x\\in S_2\\,|\\,\\pm x_2>1\/2\\}\\\\[2pt]\nS^{\\pm}_3 & := & \\{x\\in S_3\\,|\\,\\pm x_3>1\/2\\}\\\\[2pt]\nQ^+_1& := & \\{x\\in\\mathbb{R}^3\\,|\\,00$, we have the Friedrichs inequality \n\\begin{equation}\\label{PoincareModified}\n\\kappa(a)\\int_{0}^{1\/2}\\phi^2\\,dt \\le \\int_{0}^{+\\infty}(\\partial_t\\phi)^2\\,dt+a^2\\int_{1\/2}^{+\\infty}\\phi^2\\,dt,\\qquad \\forall \\phi\\in\\mrm{H}^1(0;+\\infty),\n\\end{equation}\nwhere $\\kappa(a)$ is the smallest positive root of the transcendental equation\n\\begin{equation}\\label{TransEqn}\n\\sqrt{\\kappa}\\tan\\bigg(\\cfrac{\\sqrt{\\kappa}}{2}\\bigg)=a.\n\\end{equation}\nIn particular, solving (\\ref{TransEqn}) with $a=\\pi\\sqrt{5\/2}$, we find $\\kappa(a)>\\pi^2\/2$. Therefore, using (\\ref{PoincareModified}), we can write\n\\begin{equation}\\label{PoincareTranche01}\n\\cfrac{\\pi^2}{2}\\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{S_2}(\\partial_{x_2}\\varphi)^2\\,dx+\\cfrac{5\\pi^2}{2}\\int_{S_2^+\\cup S_2^-}\\varphi^2\\,dx\n\\end{equation}\nas well as \n\\begin{equation}\\label{PoincareTranche02}\n\\cfrac{\\pi^2}{2}\\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{S_3}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{5\\pi^2}{2}\\int_{S_3^+\\cup S_3^-}\\varphi^2\\,dx.\n\\end{equation}\nThen inserting (\\ref{PoincareTranche}) in (\\ref{PoincareTranche01}), (\\ref{PoincareTranche02}) and summing up the resulting estimates, we obtain\n\\begin{equation}\\label{PoincareTranche1}\n\\begin{array}{rcl}\n\\pi^2\\displaystyle\\int_{Q^+_1}\\varphi^2\\,dx &\\le&\\displaystyle \\int_{S_2}(\\partial_{x_2}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_2^+\\cup S_2^-}(\\partial_{x_3}\\varphi)^2\\,dx+\\displaystyle \\int_{S_3}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_3^+\\cup S_3^-}(\\partial_{x_2}\\varphi)^2\\,dx\\\\[10pt]\n & & +\\displaystyle\\cfrac{1}{2}\\int_{S_2^+\\cup S_2^-\\cup S_3^+\\cup S_3^-}(\\partial_{x_1}\\varphi)^2\\,dx.\n\\end{array}\n\\end{equation}\nOn the other hand, (\\ref{PoincareTranche}) also yields\n\\begin{equation}\\label{PoincareTranche2}\n2\\pi^2\\int_{S^+_2\\cup S^-_2\\cup S^+_3\\cup S^-_3}\\varphi^2\\,dx \\le \\cfrac{1}{2} \\int_{S^+_2\\cup S^-_2\\cup S^+_3\\cup S^-_3}(\\partial_{x_1}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_2^+\\cup S_2^-}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_3^+\\cup S_3^-}(\\partial_{x_2}\\varphi)^2\\,dx.\n\\end{equation}\nFinally, summing up (\\ref{PoincareTranche0}), (\\ref{PoincareTranche0bis}), (\\ref{PoincareTranche1}) and (\\ref{PoincareTranche2}), we get \n\\[\n2\\pi^2 \\int_{\\Omega^+_1}\\varphi^2\\,dx \\le \\int_{\\Omega^+_1}|\\nabla\\varphi|^2\\,dx.\n\\]\nThis estimate together with the identity (\\ref{IdentityEigen}) imply $\\varphi\\equiv0$.\n\\end{proof}\n\\noindent We can now establish the main result of this section.\\\\[10pt]\n\\textit{Proof of Theorem \\ref{ThmDiscreteSpectrum}.} From Proposition \\ref{PropoAtLeastOne}, we know that there is at least one eigenvalue $\\mu_1$ of $A^{\\Omega}$ below $2\\pi^2$. Assume that $A^{\\Omega}$ has a second eigenvalue $\\mu_2$ such that $\\mu_2<2\\pi^2$. Set $\\Omega^+:=\\{x\\in\\Omega\\,|\\,x_1>0,\\,x_2>0,\\,x_3>0\\}$ (see Figure \\ref{ExplodedDomain} right). Then according to the result of symmetry of Lemma \\ref{LemmaSym}, the problem \n\\[\n\\begin{array}{|rcll}\n-\\Delta v &=&\\mu v &\\mbox{in }\\Omega^+\\\\[3pt]\nv&=&0 &\\mbox{on }\\Sigma_0:=\\partial\\Omega\\cap \\partial \\Omega^+\\\\[3pt]\n\\partial_nv&=&0 &\\mbox{on }\\partial \\Omega^+\\setminus\\Sigma_0\n\\end{array}\n\\]\nadmits the two eigenvalues $\\mu_1$, $\\mu_2$. Besides, from the max-min principle (\\cite[Thm. 10.2.2]{BiSo87}), we have\n\\[\n\\mu_2=\\max_{E\\subset\\mathscr{E}_1}\\inf_{w\\in E\\setminus\\{0\\}}\\cfrac{\\displaystyle\\int_{\\Omega^+}|\\nabla w|^2\\,dx}{\\displaystyle\\int_{\\Omega^+} w^2\\,dx}\\,\n\\]\nwhere $\\mathscr{E}_1$ denotes the set of subspaces of $\\mrm{H}^1_{0}(\\Omega^+;\\Sigma_0):=\\{\\varphi\\in\\mrm{H}^1(\\Omega^+)\\,|\\,\\varphi=0\\mbox{ on }\\Sigma_0\\}$ of codimension one. In particular, we have \n\\begin{equation}\\label{Contra1}\n\\mu_2 \\ge \\inf_{w\\in E\\setminus\\{0\\}}\\cfrac{\\displaystyle\\int_{\\Omega^+}|\\nabla w|^2\\,dx}{\\displaystyle\\int_{\\Omega^+} w^2\\,dx}\n\\end{equation}\nwith $E=\\{\\varphi\\in\\mrm{H}^1_{0}(\\Omega^+;\\Sigma_0)\\,|\\,\\int_{Q^+}\\varphi\\,dx=0\\}$, $Q^+:=(0;1\/2)^3$. However from the Poincar\\'e inequality, we can write for $w\\in E\\setminus\\{0\\}$, \n\\begin{equation}\\label{IdentityEigenD}\n2\\pi^2\\int_{\\Omega^+\\setminus\\overline{Q^+}}w^2\\,dx \\le \\int_{\\Omega^+\\setminus\\overline{Q^+}} |\\nabla w|^2\\,dx\n\\end{equation}\nand there holds, according to the max-min principle, \n\\begin{equation}\\label{Contra2}\n\\inf_{w\\in E\\setminus\\{0\\}}\\cfrac{\\displaystyle\\int_{Q^+}|\\nabla w|^2\\,dx}{\\displaystyle\\int_{Q^+} w^2\\,dx}=4\\pi^2\n\\end{equation}\nbecause the first positive eigenvalue of the Neumann Laplacian in $Q^+$ is equal to $4\\pi^2$. Using (\\ref{IdentityEigenD}) and (\\ref{Contra2}) in (\\ref{Contra1}) leads to $\\mu_2\\ge2\\pi$ which contradicts the initial assumption. Therefore $A^{\\Omega}$ cannot have two eigenvalues below the continuous spectrum.\\hfill $\\square$\n\n\n\\section{Absence of threshold resonance}\\label{sectionAbsenceofBoundedSol}\n\nIn this section, we establish Theorem \\ref{ThmNoBoundedSol}. To proceed, we apply the tools of \\cite{Pank17,BaNa21} that we recall now. For $R\\ge1\/2$ and $j=1,2,3$, define the truncated cylinder\n\\[\nL^R_j := \\{x\\in L_j\\,|\\,|x_j|2\\pi^2$.\\\\\n\\newline\nTherefore from now our goal is to show that the second eigenvalue of $B_R$ is larger than $2\\pi^2$ (Proposition \\ref{PropoBoundedSecond} below). We start with a result of symmetry similar to Lemma \\ref{LemmaSym}. \n\n\n\\begin{lemma}\\label{LemmaSymR}\nFor $R$ large enough, if $v\\in\\mrm{H}^1_{0}(\\Omega^R;\\partial\\Omega^R\\cap\\partial\\Omega)$ is an eigenfunction of the operator $B^R$ associated with an eigenvalue $\\mu\\le2\\pi^2$, then $v$ is symmetric with respect to the planes $(Ox_2x_3)$, $(Ox_3x_1)$, $(Ox_1x_2)$.\n\\end{lemma}\n\\begin{proof}\nThe demonstration follows the lines of the one of Lemma \\ref{LemmaSym}. However we write the details for the sake of clarity. We focus our attention on the symmetry with respect to the plane $(Ox_2x_3)$, the two other ones being similar. Introduce the function $\\varphi$ such that\n\\[\n\\varphi(x_1,x_2,x_3)=v(x_1,x_2,x_3)-v(-x_1,x_2,x_3).\n\\]\nWe wish to show that $\\varphi\\equiv0$. Set $\\Omega^{R+}_1:=\\{x\\in\\Omega^R\\,|\\,x_1>0\\}$. We have \n\\begin{equation}\\label{IdentityEigenR}\n\\int_{\\Omega^{R+}_1}|\\nabla\\varphi|^2\\,dx=\\mu\\int_{\\Omega^{R+}_1}\\varphi^2\\,dx.\n\\end{equation}\nDefine the domains \n\\[\n\\begin{array}{rcl}\nL^{R+}_1 & := & \\{x\\in L^R_1\\,|\\,x_1>1\/2\\}\\\\[2pt]\nS^R_2 & := & \\{x\\in L^R_2\\,|\\,x_1>0\\}=\\Omega^{R+}_1\\cap L^R_2\\\\[2pt]\nS^R_3 & := & \\{x\\in L^R_3\\,|\\,x_1>0\\}=\\Omega^{R+}_1\\cap L^R_3\\\\[2pt]\nS^{R\\pm}_2 & := & \\{x\\in S^R_2\\,|\\,\\pm x_2>1\/2\\}\\\\[2pt]\nS^{R\\pm}_3 & := & \\{x\\in S^R_3\\,|\\,\\pm x_3>1\/2\\}.\n\\end{array}\n\\]\nIn $L^{R+}_1$ the Poincar\\'e inequality gives\n\\begin{equation}\\label{PoincareTranche0R}\n2\\pi^2 \\int_{L^{R+}_1}\\varphi^2\\,dx \\le \\int_{L^{R+}_1}|\\nabla\\varphi|^2\\,dx.\n\\end{equation}\nOn the other hand, in $Q^+_1=(0;1\/2)\\times(-1\/2;1\/2)^2$ there holds\n\\begin{equation}\\label{PoincareTranche0bisR}\n\\pi^2 \\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{Q^+_1}(\\partial_{x_1}\\varphi)^2\\,dx.\n\\end{equation}\nThe Lemma \\ref{LemmaPoincareFriedrichsR} in Appendix (see also Lemma 5.1 of \\cite{BaMaNa17}) guarantees that for $R>1\/2$ and $a>0$, we have\n\\begin{equation}\\label{PoincareModifiedR}\n \\kappa(a,R)\\int_{0}^{1\/2}\\phi^2\\,dt \\le \\int_{0}^{R}(\\partial_t\\phi)^2\\,dt+a^2\\int_{1\/2}^{R}\\phi^2\\,dt,\\qquad \\forall\\phi\\in\\mrm{H}^1(0;R),\n\\end{equation}\nwhere $\\kappa(a,R)>0$ converges to the constant $\\kappa(a)$ appearing in (\\ref{TransEqn}) as $R\\to+\\infty$. For $a=\\pi\\sqrt{5\/2}$, as already said, one finds $\\kappa(a)>\\pi^2\/2$. Introduce $\\delta>0$ such that $\\kappa(\\pi\\sqrt{5\/2})>\\pi^2\/2+\\delta$. We know that there is $R_0$ large enough such that we have $\\kappa(\\pi\\sqrt{5\/2},R)>\\pi^2\/2+\\delta\/2$ for all $R\\ge R_0$. We infer that we have\n\\begin{equation}\\label{PoincareTranche01R}\n\\cfrac{\\pi^2+\\delta}{2}\\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{S^R_2}(\\partial_{x_2}\\varphi)^2\\,dx+\\cfrac{5\\pi^2}{2}\\int_{S_2^{R+}\\cup S_2^{R-}}\\varphi^2\\,dx\n\\end{equation}\nand\n\\begin{equation}\\label{PoincareTranche02R}\n\\cfrac{\\pi^2+\\delta}{2}\\int_{Q^+_1}\\varphi^2\\,dx \\le \\int_{S^R_3}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{5\\pi^2}{2}\\int_{S_3^{R+}\\cup S_3^{R-}}\\varphi^2\\,dx.\n\\end{equation}\nBut from the Poincar\\'e inequality in the transverse section of $S_2^{R\\pm}$, $S_3^{R\\pm}$, we know that \n\\begin{equation}\\label{PoincareTrancheR}\n5\\pi^2\\int_{S_2^{R\\pm}}\\varphi^2\\,dx \\le \\int_{S_2^{R\\pm}}(\\partial_{x_1}\\varphi)^2+(\\partial_{x_3}\\varphi)^2\\,dx,\\qquad 5\\pi^2\\int_{S_3^{R\\pm}}\\varphi^2\\,dx \\le \\int_{S_3^{R\\pm}}(\\partial_{x_1}\\varphi)^2+(\\partial_{x_2}\\varphi)^2\\,dx.\n\\end{equation}\nInserting (\\ref{PoincareTrancheR}) in (\\ref{PoincareTranche01R}), (\\ref{PoincareTranche02R}) and summing up the resulting estimates, we obtain\n\\begin{equation}\\label{PoincareTranche1R}\n\\begin{array}{rcl}\n(\\pi^2+\\delta)\\displaystyle\\int_{Q^+_1}\\varphi^2\\,dx &\\le&\\displaystyle \\int_{S^R_2}(\\partial_{x_2}\\varphi)^2\\,dx+\\displaystyle \\int_{S_3^R}(\\partial_{x_3}\\varphi)^2\\,dx\\\\[10pt]\n & &+\\displaystyle\\cfrac{1}{2}\\int_{S_2^{R+}\\cup S_2^{R-}}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_3^{R+}\\cup S_3^{R-}}(\\partial_{x_2}\\varphi)^2\\,dx\\\\[10pt]\n & & +\\displaystyle\\cfrac{1}{2}\\int_{S_2^{R+}\\cup S_2^{R-}\\cup S_3^{R+}\\cup S_3^{R-}}(\\partial_{x_1}\\varphi)^2\\,dx.\n\\end{array}\n\\end{equation}\nOn the other hand, (\\ref{PoincareTrancheR}) also yields\n\\begin{equation}\\label{PoincareTranche2R}\n\\begin{array}{l}\n\\dsp2\\pi^2\\int_{S^{R+}_2\\cup S^{R-}_2\\cup S^{R+}_3\\cup S^{R-}_3}\\varphi^2\\,dx \\\\[10pt]\n\\displaystyle\\le \\cfrac{1}{2} \\int_{S^{R+}_2\\cup S^{R-}_2\\cup S^{R+}_3\\cup S^{R-}_3}(\\partial_{x_1}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_2^{R+}\\cup S_2^{R-}}(\\partial_{x_3}\\varphi)^2\\,dx+\\cfrac{1}{2}\\int_{S_3^{R+}\\cup S_3^{R-}}(\\partial_{x_2}\\varphi)^2\\,dx.\n\\end{array}\n\\end{equation}\nFinally, summing up (\\ref{PoincareTranche0R}), (\\ref{PoincareTranche0bisR}), (\\ref{PoincareTranche1R}) and (\\ref{PoincareTranche2R}), we get \n\\[\n\\delta\\displaystyle\\int_{Q^+_1}\\varphi^2\\,dx+2\\pi^2 \\int_{\\Omega^{R+}_1}\\varphi^2\\,dx \\le \\int_{\\Omega^{R+}_1}|\\nabla\\varphi|^2\\,dx.\n\\]\nThis estimate together with the identity (\\ref{IdentityEigenR}) imply $\\int_{Q^+_1}\\varphi^2\\,dx=0$ and so $\\varphi\\equiv0$ in $Q^+_1$. From the unique continuation principle, this gives $\\varphi\\equiv0$ in $\\Omega^{R+}_1$.\n\\end{proof}\n\n\\begin{proposition}\\label{PropoBoundedSecond}\nFor $R$ large enough, the second eigenvalue $\\mu_2^R$ of $B^R$ satisfies $\\mu_2^R>2\\pi^2$. \n\\end{proposition}\n\\begin{proof}\nOne applies Lemma \\ref{LemmaSymR} to reduce the analysis to $\\Omega^{R+}:=\\{x\\in\\Omega^R\\,|\\,x_1>0,\\,x_2>0,\\,x_3>0\\}$. Here in particular we use the assumption that $R$ is large enough. The rest of the proof is completely similar to the one of Theorem \\ref{ThmDiscreteSpectrum}. \n\\end{proof}\n\n\\section{Model problems for the spectral bands}\\label{SectionModels}\n\nIn this section, we establish Theorem \\ref{MainThmPerio} and obtain models for the spectral bands $\\Upsilon^\\varepsilon_p$ appearing in (\\ref{SpectralBands}). We recall that the spectrum of $A^\\varepsilon$ is such that $\\sigma(A^\\varepsilon)=\\bigcup_{p\\in\\mathbb{N}^\\ast}\\Upsilon^\\varepsilon_p$.\n\n\\subsection{Study of $\\Upsilon^\\varepsilon_1$}\nBy definition, we have \n\\begin{equation}\\label{Def_FirstBande}\n\\Upsilon^\\varepsilon_1=\\{\\Lambda_1^\\varepsilon(\\eta),\\,\\eta\\in[0;2\\pi)^3\\}\n\\end{equation}\nwhere $\\Lambda_1^\\varepsilon(\\eta)$ is the first eigenvalue of (\\ref{PbSpectralCell}). Therefore our goal is to obtain an asymptotic expansion of $\\Lambda_1^\\varepsilon(\\eta)$ with respect to $\\varepsilon$ as $\\varepsilon\\to0^+$.\\\\\n\\newline\nPick $\\eta\\in[0;2\\pi)^3$. Let $u^\\varepsilon(\\cdot,\\eta)$ be an eigenfunction associated with $\\Lambda_1^\\varepsilon(\\eta)$. As a first approximation when $\\varepsilon\\to0^+$, it is natural to consider the expansion\n\\begin{equation}\\label{ExpansionTR0}\n\\Lambda_1^\\varepsilon(\\eta)=\\varepsilon^{-2}\\mu_1+\\dots,\\qquad u^\\varepsilon(x,\\eta)=v_1(x_1\/\\varepsilon)+\\dots\n\\end{equation}\nwhere $\\mu_1\\in(0;2\\pi^2)$ stands for the eigenvalue of the discrete spectrum of the operator $A^{\\Omega}$ introduced in Theorem \\ref{ThmDiscreteSpectrum} and $v_1\\in\\mrm{H}^1_0(\\Omega)$ is a corresponding eigenfunction that we choose such that $\\|v_1\\|_{\\mrm{L}^2(\\Omega)}=1$. Indeed, inserting $(\\varepsilon^{-2}\\mu_1,v_1(\\cdot\/\\varepsilon))$ in Problem (\\ref{PbSpectralCell}) only leaves a small discrepancy on the square faces of $\\partial\\omega^\\varepsilon$ because $v_1$ is exponentially decaying at infinity. Let us write more precisely the decomposition of $v_1$ at infinity because this will be useful in the sequel. To proceed and keep short notation, we shall work with the coordinates $(z_j,y_j)$, $j=1,\\dots,6$, such that\n\\begin{equation}\\label{Def_coor_local}\n\\begin{array}{|l}\nz_1=x_1,\\quad y_1=(x_2,x_3);\\\\[3pt]\nz_2=x_2,\\quad y_2=(x_3,x_1);\\\\[3pt]\nz_3=x_3,\\quad y_3=(x_1,x_2);\n\\end{array}\\qquad\\qquad \\begin{array}{|l}\nz_4=-x_1,\\quad y_4=(x_2,-x_3);\\\\[3pt]\nz_5=-x_2,\\quad y_5=(x_3,-x_1);\\\\[3pt]\nz_6=-x_3,\\quad y_6=(x_1,-x_2).\n\\end{array}\n\\end{equation}\nWe also define for $j=1,\\dots,6$, the branch\n\\[\n\\mathcal{L}_j:=\\{x\\in\\Omega\\,|\\,z_j>1\/2\\}.\n\\]\nThen Fourier decomposition together with the result of symmetry of Lemma \\ref{LemmaSym} and the fact that $\\mu_1$ is a simple eigenvalue\\footnote{This is needed to show that $K$ is the same in the branches $\\mathcal{L}_j$, $j=1,\\dots,6$.} guarantee that for $j=1,\\dots,6$, we have \n\\[\nv_1(x)=K\\,e^{-\\beta_1 z_j}\\,U_{\\dagger}(y_j)+O(e^{-\\beta_2 z_j})\\quad\\mbox{ in }\\mathcal{L}_j\\quad\\mbox{ as } z_j\\to+\\infty.\n\\]\nHere $K\\in\\mathbb{R}^*$ is independent of $j$, $\\beta_1:=\\sqrt{2\\pi^2-\\mu_1}$, $\\beta_2:=\\sqrt{5\\pi^2-\\mu_1}$ and \n\\begin{equation}\\label{defUdagger}\nU_{\\dagger}(s_1,s_2)=2\\cos(\\pi s_1)\\cos(\\pi s_2).\n\\end{equation}\nThe first model (\\ref{ExpansionTR0}) is interesting but does not comprise the dependence with respect $\\eta$. To improve it, consider the more refined ans\\\"atze\n\\begin{equation}\\label{ExpansionTR1}\n\\Lambda_1^\\varepsilon(\\eta)=\\varepsilon^{-2}\\mu_1+\\varepsilon^{-2}e^{-\\beta_1\/\\varepsilon}M(\\eta)+\\dots,\\qquad u^\\varepsilon(x,\\eta)=v_1(x_1\/\\varepsilon)+e^{-\\beta_1\/\\varepsilon}V(x_1\/\\varepsilon,\\eta)+\\dots\n\\end{equation}\nwhere the quantities $M(\\eta)$, $V(\\cdot,\\eta)$ are to determine. Inserting (\\ref{ExpansionTR1}) into (\\ref{PbSpectralCell}), first we obtain that $V(\\cdot,\\eta)$ must satisfy\n\\begin{equation}\\label{Pb_corrector_TR}\n\\begin{array}{|rccl}\n-\\Delta V(\\cdot,\\eta)-\\mu_1V(\\cdot,\\eta)&=&M(\\eta)v_1 & \\mbox{ in }\\Omega\\\\[3pt]\n V(\\cdot,\\eta) &=&0 &\\mbox{ on }\\partial\\Omega.\n\\end{array}\n\\end{equation}\nIn order to have a non zero solution to (\\ref{Pb_corrector_TR}), we must look for a $V(\\cdot,\\eta)$ which is growing at infinity. Due to (\\ref{Pb_corrector_TR}), the simplest growth that we can allow is\n\\[\nV(x,\\eta)=B_j\\,e^{\\beta_1 z_j}\\,U_{\\dagger}(y_j)+O(e^{-\\beta_1 z_j})\\quad\\mbox{ in }\\mathcal{L}_j\\quad\\mbox{ as } z_j\\to+\\infty\n\\]\nwhere the $B_j$ are some constants. Then the quasi-periodic conditions satisfied by $u^\\varepsilon(\\cdot,\\eta)$ at the faces located at $x_1=\\pm1\/2$, $x_2=\\pm1\/2$, $x_3=\\pm1\/2$ (see (\\ref{PbSpectralCell})) lead us to choose the $B_j$ such that\n\\[\n\\begin{array}{|l}\nK+B_4=e^{i\\eta_1}(K+B_1)\\\\[2pt]\nK-B_4=e^{i\\eta_1}(-K+B_1)\n\\end{array}\\qquad \\begin{array}{|l}\nK+B_5=e^{i\\eta_2}(K+B_2)\\\\[2pt]\nK-B_5=e^{i\\eta_2}(-K+B_2)\n\\end{array}\\qquad\\begin{array}{|l}\nK+B_6=e^{i\\eta_3}(K+B_3)\\\\[2pt]\nK-B_6=e^{i\\eta_3}(-K+B_3).\n\\end{array}\n\\]\nSolving these systems, we obtain $B_j=K\\,e^{-i\\eta_j}$, $B_{j+3}=K\\,e^{+i\\eta_j}$ for $j=1,2,3$. Now since $\\mu_1$ is a simple eigenvalue of $A^\\Omega$, multiplying (\\ref{Pb_corrector_TR}) by $v_1$, integrating by part in $\\Omega^R$ and taking the limit $R\\to+\\infty$, we find that there is a solution if and only if the following compatibility condition \n\\[\nM(\\eta)\\|v_1\\|_{\\mrm{L}^2(\\Omega)}=-2\\beta_1K^2\\sum_{j=1}^3e^{i\\eta_j}+e^{-i\\eta_j}\\qquad\\Leftrightarrow\\qquad M(\\eta)=-4\\beta_1K^2\\sum_{j=1}^3\\cos(\\eta_j)\n\\]\nis satisfied. This defines the value of $M(\\eta)$ in the expansion (\\ref{ExpansionTR1}). From (\\ref{Def_FirstBande}), this gives the inclusion $\\Upsilon^\\varepsilon_1\\subset \\varepsilon^{-2}\\mu_1+\\varepsilon^{-2}e^{-\\sqrt{2\\pi^2-\\mu_1}\/\\varepsilon}(-c_1;c_1)$ for $\\varepsilon$ small enough of Theorem \\ref{MainThmPerio}. For $c_1$, we can take any constant larger than $12\\beta_1K^2$. Note that we decided to focus on a rather formal presentation above for the sake of conciseness. We emphasize that all these results can be completely justified by proving rigorous error. This has been realized in detail in \\cite{Naza17,Naza18} for similar problems and can be repeated with obvious modifications.\n\n\n\n\n\n\n\n\n\\subsection{Study $\\Upsilon^\\varepsilon_k$, $k\\ge 2$}\n\nWe turn our attention to the asymptotic of the spectral bands of higher frequency\n\\begin{equation}\\label{Def_FirstBande_2}\n\\Upsilon^\\varepsilon_k=\\{\\Lambda_k^\\varepsilon(\\eta),\\,\\eta\\in[0;2\\pi)^3\\},\\qquad k\\ge2,\n\\end{equation}\nas $\\varepsilon\\to0^+$. Pick $\\eta\\in[0;2\\pi)^3$ and introduce $u^\\varepsilon(\\cdot,\\eta)$ an eigenfunction associated with $\\Lambda_k^\\varepsilon(\\eta)$. In the sequel, to simplify, we remove the subscript ${}_k$ and do not indicate the dependence on $\\eta$. As a first approximation when $\\varepsilon\\to0^+$, we consider the expansion\n\\begin{equation}\\label{ExpansionTR0_2}\n\\Lambda^\\varepsilon=\\varepsilon^{-2}2\\pi^2+\\nu+\\dots,\\qquad u^\\varepsilon(x)=v^\\varepsilon(x)+\\dots\n\\end{equation}\nwith $v^\\varepsilon$ of the form\n\\[\nv^\\varepsilon(x)=\\begin{array}{|rl}\n\\gamma^\\pm_{1}(x_1)\\,U_{\\dagger}(x_2\/\\varepsilon,x_3\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_1:=\\{x\\in\\omega^{\\varepsilon}_1\\,|\\,\\pm x_1>\\varepsilon\/2\\},\\\\[3pt]\n\\gamma^\\pm_{2}(x_2)\\,U_{\\dagger}(x_3\/\\varepsilon,x_1\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_2:=\\{x\\in\\omega^{\\varepsilon}_2\\,|\\,\\pm x_2>\\varepsilon\/2\\},\\\\[3pt]\n\\gamma^\\pm_{3}(x_3)\\,U_{\\dagger}(x_1\/\\varepsilon,x_2\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_3:=\\{x\\in\\omega^{\\varepsilon}_3\\,|\\,\\pm x_3>\\varepsilon\/2\\},\\\\[3pt]\n\\end{array}\n\\]\nwhere the functions $\\gamma^\\pm_{j}$, $j=1,2,3$, are to determine ($U_{\\dagger}$ is defined in (\\ref{defUdagger})). Inserting (\\ref{ExpansionTR0_2}) into Problem (\\ref{PbSpectralCell}), we obtain for $j=1,2,3$,\n\\begin{equation}\\label{farfield}\n\\begin{array}{|rcl}\n\\partial^2_s\\gamma^+_{j}+\\nu\\gamma^+_{j}&=&0\\quad\\mbox{ in }(0;1\/2)\\\\[3pt]\n\\partial^2_s\\gamma^-_{j}+\\nu\\gamma^-_{j}&=&0\\quad\\mbox{ in }(-1\/2;0)\\\\[3pt]\n\\gamma^-_{j}(-1\/2)&=&e^{i\\eta_j}\\gamma^+_{j}(+1\/2)\\\\[3pt]\n\\partial_s\\gamma^-_{j}(-1\/2)&=&e^{i\\eta_j}\\partial_s\\gamma^+_{j}(+1\/2).\n\\end{array}\n\\end{equation}\nTo uniquely define the $\\gamma^\\pm_{j}$, we need to complement (\\ref{farfield}) with some conditions at the origin. To proceed, we match the behaviour of the $\\gamma^\\pm_{j}$ with the one of some inner field expansion of $u^\\varepsilon$. More precisely, in a neighbourhood of the origin we look for an expansion of $u^\\varepsilon$ of the form \n\\begin{equation}\\label{NearField}\nu^\\varepsilon(x)=W(x_1\/\\varepsilon)+\\dots\n\\end{equation}\nwith $W$ to determine. Inserting (\\ref{NearField}) and (\\ref{ExpansionTR0_2}) in (\\ref{PbSpectralCell}), we find that $W$ must satisfy \n\\begin{equation}\\label{NearFieldPb}\n\\begin{array}{|rcll}\n\\Delta W+2\\pi^2W&=&0&\\mbox{ in }\\Omega\\\\\nW&=&0 &\\mbox{ on }\\partial\\Omega.\n\\end{array}\n\\end{equation}\nNow we come to the point where Theorem \\ref{ThmNoBoundedSol} appears in the analysis. Indeed it guarantees that the only solution of (\\ref{NearFieldPb}) which is bounded at infinity is the null function. Therefore we take $W\\equiv0$ and impose \n\\begin{equation}\\label{farfield_BC}\n\\gamma^\\pm_{j}(0)=0.\n\\end{equation}\nThen solving the spectral problem (\\ref{farfield}), (\\ref{farfield_BC}), we obtain \n\\begin{equation}\\label{DefModel1D}\n\\nu=p^2\\pi^2 \\mbox{ for }p\\in\\mathbb{N}^\\ast,\\qquad \\begin{array}{|l}\n\\gamma^+_{j}(s)=\\sin(p\\pi s)\\\\[2pt]\n\\gamma^-_{j}(s)=-e^{i\\eta_j}\\sin(p\\pi s).\n\\end{array}\n\\end{equation}\nNote that $\\nu$ is a triple eigenvalue (geometric multiplicity equal to three) because the problems (\\ref{farfield}), (\\ref{farfield_BC}) for $j=1,2,3$ are uncoupled. Additionally $\\nu$ is independent of $\\eta\\in[0;2\\pi)^3$. This latter fact is not completely satisfactory and in the sequel we wish to improve the model obtained above. Let us refine the expansion proposed in (\\ref{ExpansionTR0_2}) and work with\n\\begin{equation}\\label{ExpansionTR0_3}\n\\Lambda^\\varepsilon=\\cfrac{2\\pi^2}{\\varepsilon^2}+p^2\\pi^2+\\varepsilon\\tilde{\\nu}+\\dots,\\quad u^\\varepsilon(x)=\\begin{array}{|rl}\n(a_1\\gamma^\\pm_{1}(x_1)+\\varepsilon\\tilde{\\gamma}^{\\pm}_{1}(x_1))\\,U_{\\dagger}(x_2\/\\varepsilon,x_3\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_1\\\\[3pt]\n(a_2\\gamma^\\pm_{2}(x_2)+\\varepsilon\\tilde{\\gamma}^{\\pm}_{2}(x_2))\\,U_{\\dagger}(x_3\/\\varepsilon,x_1\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_2\\\\[3pt]\n(a_3\\gamma^\\pm_{3}(x_3)+\\varepsilon\\tilde{\\gamma}^{\\pm}_{3}(x_3))\\,U_{\\dagger}(x_1\/\\varepsilon,x_2\/\\varepsilon)&\\mbox{ in }\\omega^{\\varepsilon\\pm}_3\\\\[3pt]\n\\end{array}+\\dots .\n\\end{equation}\nHere $a:=(a_1,a_2,a_3)\\in\\mathbb{R}^3$, $\\tilde{\\nu}$ as well as the $\\tilde{\\gamma}^{\\pm}_{j}$, $j=1,2,3$, are to determine. Since we are working with eigenfunctions, we can impose the normalization condition $a_1^2+a_2^2+a_3^2=1$. Inserting (\\ref{ExpansionTR0_3}) into Problem (\\ref{PbSpectralCell}) and extracting the terms in $\\varepsilon$, we get for $j=1,2,3$,\n\\begin{equation}\\label{farfield_3}\n\\begin{array}{|rcl}\n\\partial^2_s\\tilde{\\gamma}^+_{j}+p^2\\pi^2\\tilde{\\gamma}^+_{j}&=&-\\tilde{\\nu}a_j\\gamma^+_j\\quad\\mbox{ in }(0;1\/2)\\\\[3pt]\n\\partial^2_s\\tilde{\\gamma}^-_{j}+p^2\\pi^2\\tilde{\\gamma}^-_{j}&=&-\\tilde{\\nu}a_j\\gamma^-_j\\quad\\mbox{ in }(-1\/2;0)\\\\[3pt]\n\\tilde{\\gamma}^-_{j}(-1\/2)&=&e^{i\\eta_j}\\tilde{\\gamma}^+_{j}(+1\/2)\\\\[3pt]\n\\partial_s\\tilde{\\gamma}^-_{j}(-1\/2)&=&e^{i\\eta_j}\\partial_s\\tilde{\\gamma}^+_{j}(+1\/2).\n\\end{array}\n\\end{equation}\nTo define properly the $\\tilde{\\gamma}^\\pm_{j}$, we need to add to (\\ref{farfield_3}) conditions at the origin. To identify them, again we match with the behaviour of some inner field representation of $u^\\varepsilon$. In a neighbourhood of the origin we look for an expansion of $u^\\varepsilon$ of the form \n\\begin{equation}\\label{NearField_3}\nu^\\varepsilon(x)=\\varepsilon \\tilde{W}(x_1\/\\varepsilon)+\\dots.\n\\end{equation}\nInserting (\\ref{NearField_3}) and (\\ref{ExpansionTR0_3}) in (\\ref{PbSpectralCell}), we find that $\\tilde{W}$ must satisfy \n\\begin{equation}\\label{NearFieldPb_3}\n\\begin{array}{|rcll}\n\\Delta \\tilde{W}+2\\pi^2\\tilde{W}&=&0&\\mbox{ in }\\Omega\\\\\n\\tilde{W}&=&0 &\\mbox{ on }\\partial\\Omega.\n\\end{array}\n\\end{equation}\nThis problem admits solutions $\\tilde{W}_{j}$, $j=1,\\dots,6$, with the expansions\n\\begin{equation}\\label{DefPolMat1}\n\\tilde{W}_j(x)=\n\\begin{array}{|rl}\n(z_j+M_{jj})\\,U_{\\dagger}(y_j)+\\dots&\\mbox{ in }\\mathcal{L}_j\\mbox{ as }z_j\\to+\\infty \\\\[2pt]\nM_{jk}\\,U_{\\dagger}(y_k)+\\dots&\\mbox{ in }\\mathcal{L}_k\\mbox{ as }z_k\\to+\\infty,\\quad k\\ne j.\n\\end{array}\n\\end{equation}\nHere we use the coordinates $(z_j,y_j)$ introduced in (\\ref{Def_coor_local}). Note that at infinity $\\tilde{W}_j$ is growing only in the branch $\\mathcal{L}_j$. Let us explain how to show the existence of these functions (see \\cite[Chap. 4, Prop. 4.13]{NaPl94} for more details). Introduce some $\\psi\\in\\mathscr{C}^{\\infty}(\\mathbb{R})$ such that $\\psi(s)=1$ for $s>1$ and $\\psi(s)=0$ for $s\\le 1\/2$. Then for $j=1,\\dots,6$, define $f_j$ such that $f_j(x)=(\\Delta+2\\pi^2)(\\psi(z_j) \\,z_jU_{\\dagger}(y_j))$. Observe that $f_j$ is compactly supported. Then the theory of \\cite[Chap. 5]{NaPl94} together with Theorem \\ref{ThmNoBoundedSol} above ensure that there is a solution to the problem\n\\[\n\\begin{array}{|rcll}\n\\Delta \\hat{W}_j+2\\pi^2\\hat{W}_j&=&-f_j&\\mbox{ in }\\Omega\\\\\n\\hat{W}_j&=&0 &\\mbox{ on }\\partial\\Omega\n\\end{array}\n\\]\nwhich is bounded at infinity. Finally we take $\\tilde{W}_j$ such that $\\tilde{W}_j(x)=\\psi(z_j) \\,z_jU_{\\dagger}(y_j)+\\hat{W}_j(x)$. The coefficients $M_{jk}$ form the so-called polarization matrix\n\\begin{equation}\\label{def_Pola_matrix}\n\\mathbb{M}:=\\left(M_{jk}\\right)_{1\\le j,k\\le 6}\n\\end{equation}\nwhich is real and symmetric even in non symmetric geometries (see \\cite[Chap. 5, Prop. 4.13]{NaPl94}).\\\\\n\\newline\nFor the functions $\\gamma^{\\pm}_{j}$ in (\\ref{DefModel1D}), we have the Taylor expansion, as $s\\to0$,\n\\begin{equation}\\label{ExpanFarField}\n\\gamma^+_{j}(s)=0+p\\pi s+\\dots=\\varepsilon p\\pi\\,\\cfrac{s}{\\varepsilon}+\\dots,\\qquad \\gamma^-_{j}(s)=0-e^{i\\eta_j}p\\pi s+\\dots=-\\varepsilon e^{i\\eta_j} p\\pi \\,\\cfrac{s}{\\varepsilon}+\\dots.\n\\end{equation}\nComparing (\\ref{ExpanFarField}) with (\\ref{DefPolMat1}) leads us to choose $\\tilde{W}$ in the expansion $u^\\varepsilon(x)=\\varepsilon \\tilde{W}(x_1\/\\varepsilon)+\\dots$ (see (\\ref{NearField_3})) such that\n\\[\n\\tilde{W}=p\\pi \\sum_{j=1}^3a_j\\,(\\tilde{W}_j+e^{i\\eta_j}\\tilde{W}_{3+j}).\n\\]\nThis sets the constant behaviour of $\\tilde{W}$ at infinity and we now match the later with the behaviour of the $\\tilde{\\gamma}^\\pm_{j}$ at the origin to close system (\\ref{farfield_3}). This step leads us to impose \n\\begin{equation}\\label{ConditionAtZero}\n\\begin{array}{ll}\n\\displaystyle\\tilde{\\gamma}^+_1(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j1}+e^{i\\eta_j}M_{3+j1}); &\\quad\\displaystyle \\tilde{\\gamma}^-_1(0)=p\\pi \\sum_{j=1}^3 a_j\\,M_{j4}+e^{i\\eta_j}M_{3+j4});\\\\[14pt]\n\\displaystyle\\tilde{\\gamma}^+_2(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j2}+e^{i\\eta_j}M_{3+j2}); &\\quad\\displaystyle \\tilde{\\gamma}^-_2(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j5}+e^{i\\eta_j}M_{3+j5});\\\\[14pt]\n\\displaystyle\\tilde{\\gamma}^+_3(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j3}+e^{i\\eta_j}M_{3+j3}); &\\quad\\displaystyle \\tilde{\\gamma}^-_3(0)=p\\pi \\sum_{j=1}^3 a_j\\,(M_{j6}+e^{i\\eta_j}M_{3+j6}).\n\\end{array}\n\\end{equation}\nEquations (\\ref{farfield_3}), (\\ref{ConditionAtZero}) form a boundary value problem for a system of ordinary differential equations. For this problem, there is a kernel and co-cokernel. In order to have a solution, the following compatibility conditions, obtained by multiplying (\\ref{farfield_3}) by $\\overline{\\gamma^\\pm_j}$ and integrating by parts,\n\\[\n\\overline{\\partial_s\\gamma^+_j(0)}\\,\\tilde{\\gamma}^+_j(0)-\\overline{\\partial_s\\gamma^-_j(0)}\\,\\tilde{\\gamma}^-_j(0)=-\\tilde{\\nu}a_j\\left(\\int_{-1\/2}^{0}|\\gamma^-_j|^2\\,ds+\\int_{0}^{1\/2}|\\gamma^+_j|^2\\,ds\\right)\n\\]\nmust be satisfied for $j=1,2,3$ (note that we used that $\\gamma^{\\pm}_j(0)=0$ according to (\\ref{farfield_BC})). Since $\\overline{\\partial_s\\gamma^+_j(0)}=p\\pi$ and $\\overline{\\partial_s\\gamma^-_j(0)}=-p\\pi e^{-i\\eta_j}$, this gives\n\\[\n2(p\\pi)^2\\bigg(\\sum_{k=1}^3 a_k\\,(M_{kj}+e^{i\\eta_k}M_{3+kj})+e^{-i\\eta_j}\\sum_{k=1}^3 a_k\\,(M_{k3+j}+e^{i\\eta_k}M_{3+k3+j})\\bigg)=-\\tilde{\\nu}a_j.\n\\]\nIn a more compact form and by rewriting the dependence with respect to $\\eta$, we obtain\n\\begin{equation}\\label{CompactForm}\n\\mathbb{A}(\\eta) a^{\\top}=-\\tilde{\\nu}(\\eta)a^{\\top},\n\\end{equation}\nwith \n\\[\n\\mathbb{A}(\\eta):=\\Theta(\\eta)\\mathbb{M}\\Theta^\\ast(\\eta)\\in\\mathbb{C}^{3\\times 3},\\quad\n\\Theta(\\eta):=\\left(\\begin{array}{cccccc}\n1 & 0 & 0 & e^{-i\\eta_1} & 0 & 0\\\\[2pt]\n0& 1 & 0 & 0 & e^{-i\\eta_2} & 0\\\\[2pt]\n0 & 0 & 1 & 0 & 0 & e^{-i\\eta_3} \n\\end{array}\\right),\\quad\\Theta^{\\ast}(\\eta)=\\overline{\\Theta(\\eta)}^{\\top}.\n\\]\nAbove we used that the polarization matrix $\\mathbb{M}$ defined in (\\ref{def_Pola_matrix}) is symmetric. By solving the spectral problem (\\ref{CompactForm}), we get the values for $\\tilde{\\nu}(\\eta)$ and $a$. Once $\\tilde{\\nu}(\\eta)$ and $a$ are known, one can compute the solution to the system (\\ref{farfield_3}), (\\ref{ConditionAtZero}) to obtain the expressions of the $\\tilde{\\gamma}_j(\\eta)$. This ends the definition of the terms appearing in the expansions (\\ref{ExpansionTR0_3}). Let us exploit and comment these results.\\\\ \n\\newline\n$\\star$ First, observe that the $\\tilde{\\nu}(\\eta)$ are real. Indeed, since $\\mathbb{M}$ is real and symmetric, we infer that $\\mathbb{A}(\\eta)$ is hermitian. \\\\\n\\newline\n$\\star$ We have obtained \n\\begin{equation}\\label{ExpanConclu}\n\\Lambda^\\varepsilon(\\eta)=\\varepsilon^{-2}2\\pi^2+p^2\\pi^2+\\varepsilon\\tilde{\\nu}(\\eta)+\\dots\\,.\n\\end{equation}\nNote that in this expansion, the third term, contrary to the first two ones, depends on $\\eta$. For $\\eta\\in[0;2\\pi)^3$, denote by $\\tilde{\\nu}_1(\\eta)$, $\\tilde{\\nu}_2(\\eta)$, $\\tilde{\\nu}_3(\\eta)$ the three eigenvalues of \n(\\ref{CompactForm}) numbered such that\n\\[\n\\tilde{\\nu}_1(\\eta) \\le \\tilde{\\nu}_2(\\eta) \\le \\tilde{\\nu}_3(\\eta).\n\\]\nDefine the quantity \n\\begin{equation}\\label{defAleph}\n\\aleph:=\\bigcup_{j=1}^3\\{\\tilde{\\nu}_j(\\eta),\\,\\eta\\in[0;2\\pi)^3\\}.\n\\end{equation}\nSince $\\Upsilon^\\varepsilon_{1+q+3(p-1)}=\\{\\Lambda_{1+q+3(p-1)}^\\varepsilon(\\eta),\\,\\eta\\in[0;2\\pi)^3\\}$, this analysis shows that we have the inclusion\n\\begin{equation}\\label{MainInclusion}\n\\Upsilon^\\varepsilon_{1+q+3(p-1)}\\subset \\varepsilon^{-2}2\\pi^2+p^2\\pi^2+\\varepsilon\\,(-c_p;c_p)\n\\end{equation}\nfor $\\varepsilon$ small enough as stated in Theorem \\ref{MainThmPerio}. For $c_p$, we can take any value such that $\\overline{\\aleph}\\subset (-c_p;c_p)$. \\\\\n\\newline\n$\\star$ Due to the symmetries of $\\Omega$, $\\mathbb{M}$ is of the form\n\\begin{equation}\\label{MatPolSimple}\n\\left(\\begin{array}{cccccc}\nr_m & t^{\\perp}_m & t^{\\perp}_m & t_m & t^{\\perp}_m & t^{\\perp}_m \\\\[2pt]\nt^{\\perp}_m & r_m & t^{\\perp}_m & t^{\\perp}_m & t_m & t^{\\perp}_m \\\\[2pt]\nt^{\\perp}_m & t^{\\perp}_m & r_m & t^{\\perp}_m & t^{\\perp}_m & t_m \\\\[2pt]\nt_m & t^{\\perp}_m & t^{\\perp}_m & r_m & t^{\\perp}_m & t^{\\perp}_m \\\\[2pt]\nt^{\\perp}_m & t_m & t^{\\perp}_m & t^{\\perp}_m & r_m & t^{\\perp}_m \\\\[2pt]\nt^{\\perp}_m & t^{\\perp}_m & t_m & t^{\\perp}_m & t^{\\perp}_m & r_m \n\\end{array}\\right)\n\\end{equation}\nwhere $r_m$, $t_m$, $t_m^\\perp$ are real coefficients. For $\\eta=(0,0,0)$, we find that the eigenvalues of (\\ref{CompactForm}) are \n\\[\n2r_m+2t_m+8t^{\\perp}_m,\\qquad 2r_m+2t_m-4t^{\\perp}_m,\\qquad 2r_m+2t_m-4t^{\\perp}_m.\n\\] \nFor $\\eta=(\\pi,\\pi,\\pi)$, we obtain \n\\[\n2r_m-2t_m,\\qquad 2r_m-2t_m,\\qquad 2r_m-2t_m.\n\\]\nIn the numerics of \\S\\ref{Section_PolMat}, we compute $\\mathbb{M}$. From the values of $r_m$, $t_m$, $t^{\\perp}_m$ obtained in (\\ref{ValuePolMat}), we find for example\n\\[\n2r_m+2t_m+8t^{\\perp}_m \\ne 2r_m-2t_m.\n\\]\nThis shows that the set $\\aleph$ appearing in (\\ref{defAleph}) has a non empty interior. Additionally, since $\\nu_1(\\pi)=\\nu_2(\\pi)=\\nu_3(\\pi)$ and the $\\nu_j$ depend continuously on $\\eta$, we deduce that $\\aleph$ is a connected segment. This is obtained in the numerics of \\S\\ref{Section_PolMat} (see (\\ref{NumAleph})) and due to the symmetries of $\\omega^\\varepsilon$, this was somehow expected.\\\\\n\\newline\nFinally, let us mention again that all the formal presentation above can be justified rigorously by a direct adaptation of the proofs of error estimates presented in \\cite{Naza17,Naza18}.\n\n\n\n\\section{Numerics}\\label{SectionNumerics}\n\n\\subsection{Spectrum of $A^{\\Omega}$}\n\nWe start the numerics by computing the spectrum of the operator $A^{\\Omega}$ defined after (\\ref{PbSpectralZoom}). More precisely, in order to approximate also the eigenvalues which are embedded in the continuous spectrum of $A^{\\Omega}$ and to reveal complex resonances which would be located close to the real axis, we work with Perfectly Matched Layers \\cite{Bera94,KiPa10,BoCP18} (see also the techniques of analytic dilatations \\cite{aguilar1971class,balslev1971spectral,moiseyev1998quantum}). For $\\theta\\in(0;\\pi\/2)$ and $L>1\/2$, define the complex valued parameters \n\\[\n\\alpha^\\theta_1=\\begin {array}{|ll}\n1 & \\mbox{for }|x_1|\\le L\\\\[2pt]\ne^{-i\\theta} & \\mbox{for }|x_1|>L\n\\end{array},\\qquad \\alpha^\\theta_2=\\begin {array}{|ll}\n1 & \\mbox{for }|x_2|\\le L\\\\[2pt]\ne^{-i\\theta} & \\mbox{for }|x_2|>L\n\\end{array},\\qquad \\alpha^\\theta_3=\\begin {array}{|ll}\n1 & \\mbox{for }|x_3|\\le L\\\\[2pt]\ne^{-i\\theta} & \\mbox{for }|x_3|>L.\n\\end{array}\n\\]\nHere the coefficient $\\theta$ will drive the rotation of the continuous spectrum while $L$ marks the beginning of the PML region. Then consider the spectral problem\n\\begin{equation}\\label{ProblemPMLs}\n\\begin{array}{|rcll}\n-\\alpha^\\theta_1\\cfrac{\\partial}{\\partial x_1}\\Big(\\alpha^\\theta_1\\cfrac{\\partial u}{\\partial x_1}\\,\\Big)+\\alpha^\\theta_2\\cfrac{\\partial}{\\partial x_2}\\Big(\\alpha^\\theta_2\\cfrac{\\partial u}{\\partial x_2}\\,\\Big)+\\alpha^\\theta_3\\cfrac{\\partial}{\\partial x_3}\\Big(\\alpha^\\theta_3\\cfrac{\\partial u}{\\partial x_3}\\,\\Big)&=&\\Lambda u &\\mbox{ in } \\Omega\\\\[3pt]\nu&=&0&\\mbox{ on } \\partial\\Omega.\n\\end{array}\n\\end{equation}\nDenote by $A^\\Omega_\\theta$ the unbounded operator associated with (\\ref{ProblemPMLs}). Observe that $A^\\Omega_\\theta$ is not selfadjoint due to the complex parameters $\\alpha^\\theta_j$. However the theory guarantees that the real eigenvalues of $A^\\Omega_\\theta$ coincide exactly with the real eigenvalues of $A^{\\Omega}$. What is interesting is that one can show that the essential spectrum of $A^\\Omega_\\theta$, that is the values of $\\Lambda$ such that $A^\\Omega_\\theta-\\Lambda\\,\\mrm{Id}:\\mathcal{D}(A^\\Omega_\\theta)\\to\\mrm{L}^2(\\Omega)$ is not Fredholm, corresponds to the set\n\\[\n\\bigcup_{m,n\\in\\mathbb{N}^\\ast}\\{\\pi^2(m^2+n^2)+t\\,e^{-2i\\theta},\\,t\\ge0\\},\n\\]\nso that the real eigenvalues of $A^\\Omega_\\theta$ are isolated in the spectrum. As a consequence, we can compute them by truncating the branches of $\\Omega$ at a certain distance without producing spectral pollution. Then we approximate the spectrum in this bounded geometry by using a classical P1 finite element method. We construct the matrices with the library \\texttt{Freefem++} \\cite{Hech12} and compute the spectrum with \\texttt{Matlab}\\footnote{\\textit{Matlab}, \\url{http:\/\/www.mathworks.com\/}.}.\\\\\n\\newline\nIn Figure \\ref{FigSpectrum}, we display in the complex plane the approximation of the spectrum of $A^\\Omega_\\theta$ obtained with this approach. We observe that the branches of essential spectrum of $A^\\Omega_\\theta$ are somehow discretized. This is due to the fact that the approximated problem is set in finite dimension. Moreover, we note that $A^\\Omega_\\theta$, and so $A^\\Omega$, have eigenvalues on the real line. In accordance with Theorem \\ref{ThmDiscreteSpectrum}, we find exactly one eigenvalue $\\mu_1\\approx 12.9\\approx1.3\\pi^2$ on the segment $(0;\\pi^2)$. We also note the presence of eigenvalues embedded in the continuous spectrum for the operator $A^{\\Omega}$. For the first one, we get $\\mu_2\\approx 46.7\\approx4.7\\pi^2$. \n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=10cm]{Spectrum.pdf}\n\\caption{Approximation of the spectrum of $A^\\Omega_\\theta$ for $\\theta=\\pi\/4$ ($L=1\/2$).}\n\\label{FigSpectrum}\n\\end{figure}\n\n\\noindent In Figure \\ref{FigEigenfunctions}, we represent an eigenfunction (trapped mode) associated with the eigenvalue $\\mu_1$ of the discrete spectrum of $A^{\\Omega}$. As guaranteed by Lemma \\ref{LemmaSym}, we observe that it is indeed symmetric with respect to the planes $(Ox_2x_3)$, $(Ox_3x_1)$, $(Ox_1x_2)$.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=5.8cm]{TMx0}\\qquad\\quad\\includegraphics[width=5.8cm]{TMy0}\n\\caption{Eigenfunction associated with the eigenvalue $\\mu_1$: cuts $x_1=0$ (left) and $x_2=0$ (right).}\n\\label{FigEigenfunctions}\n\\end{figure}\n\n\\subsection{Polarization matrix and threshold scattering matrix for the operator $A^\\Omega$}\\label{Section_PolMat}\nIn this section, we explain how to compute the polarization matrix $\\mathbb{M}$ introduced after (\\ref{CompactForm}) and whose properties allow one to assess the second corrector term in the expansion $\\Lambda^\\varepsilon=\\varepsilon^{-2}2\\pi^2+p^2\\pi^2+\\varepsilon\\tilde{\\nu}+\\dots$ of the eigenvalues generating the spectral bands $\\Upsilon^\\varepsilon_k$, $k\\ge 2$. We work in the geometry $\\Omega$ defined in (\\ref{DefOmega}) and study the problem (\\ref{PbSpectralZoom}) at the threshold, namely \n\\begin{equation}\\label{PbSpiderThreshold}\n\\begin{array}{|rcll}\n\\Delta v +2\\pi^2v&=& 0 &\\mbox{on }\\Omega\\\\\nv&=&0 &\\mbox{on }\\partial\\Omega.\n\\end{array}\n\\end{equation}\nTo obtain $\\mathbb{M}$, we will first compute the so-called threshold scattering matrix that we define now. In $\\mathcal{L}_j$, $j=1,\\dots,6$, set \n\\[\nw_{j}^{\\pm}(x)=\\cfrac{z_j \\mp i}{\\sqrt{2}}\\,U_{\\dagger}(y_j).\n\\]\nLet us work again with the function $\\psi\\in\\mathscr{C}^{\\infty}(\\mathbb{R})$ introduced after (\\ref{DefPolMat1}) such that $\\psi(s)=1$ for $s>1$ and $\\psi(s)=0$ for $s\\le 1\/2$. For $j=1,\\dots,6$, define $\\psi_j$ such that $\\psi_j(x)=\\psi(z_j,y_j)$ (observe that $\\psi_j$ is non zero only in the branch $\\mathcal{L}_j$). For $j=1,\\dots,6$, the theory of \\cite[Chap. 5]{NaPl94} guarantees that problem (\\ref{PbSpiderThreshold}) admits a solution with the decomposition\n\\begin{equation}\\label{decompo_vj}\nv_j=\\psi_j\\,w_j^{-}+\\sum_{k=1}^6\\psi_k\\,s_{jk}\\,w_k^++\\tilde{u}_j,\n\\end{equation}\nwhere the $s_{jk}$ are complex numbers and the $\\tilde{u}_j$ decay exponentially at infinity. The matrix \n\\[\n\\mathbb{S}:=\\left(s_{jk}\\right)_{1\\le j,k\\le 6}\n\\]\nis called the threshold scattering matrix. It is symmetric ($\\mathbb{S}=\\mathbb{S}^{\\top}$) but not necessarily hermitian and unitary ($\\mathbb{S}\\,\\overline{\\mathbb{S}}^{\\top}=\\mrm{Id}$). It is known (see relation (7.9) in \\cite{Naza16}) that $\\mathbb{M}$ coincides with the Cayley transform of $\\mathbb{S}$, i.e. we have\n\\begin{equation}\\label{DefMatPol}\n\\mathbb{M}=i(\\mrm{Id}+\\mathbb{S})^{-1}(\\mrm{Id}-\\mathbb{S}).\n\\end{equation}\nNote that one can show that we have $\\dim\\,\\ker\\,(\\mrm{Id}+\\mathbb{S})=\\dim\\,(\\mathscr{B}\/\\mathscr{B}_{\\mrm{tr}})$ (the quotient space) where $\\mathscr{B}$ denotes the space of bounded solutions of (\\ref{PbSpiderThreshold}) and $\\mathscr{B}_{\\mrm{tr}}$ the space of trapped modes of (\\ref{PbSpiderThreshold}). For the proof, we refer the reader for example to Theorem 1 in \\cite{Naza14}. Since Theorem \\ref{ThmNoBoundedSol} ensures that $\\mathscr{B}$ reduces to the null function, we infer that $\\mrm{Id}+\\mathbb{S}$ is invertible which guarantees that $\\mathbb{M}$ is well defined via formula (\\ref{DefMatPol}). Additionally, due to the symmetries of $\\Omega$, $\\mathbb{S}$ is of the form\n\\begin{equation}\\label{DefMatScaSimplifiee}\n\\left(\\begin{array}{cccccc}\nr & t^{\\perp} & t^{\\perp} & t & t^{\\perp} & t^{\\perp} \\\\[2pt]\nt^{\\perp} & r & t^{\\perp} & t^{\\perp} & t & t^{\\perp} \\\\[2pt]\nt^{\\perp} & t^{\\perp} & r & t^{\\perp} & t^{\\perp} & t \\\\[2pt]\nt & t^{\\perp} & t^{\\perp} & r & t^{\\perp} & t^{\\perp} \\\\[2pt]\nt^{\\perp} & t & t^{\\perp} & t^{\\perp} & r & t^{\\perp} \\\\[2pt]\nt^{\\perp} & t^{\\perp} & t & t^{\\perp} & t^{\\perp} & r \n\\end{array}\\right)\n\\end{equation}\nwhere $r$, $t$, $t^{\\perp}$ are complex reflection and transmission coefficients. Therefore it is sufficient to compute $v_1$. To proceed, we shall work in the bounded domain $\\Omega^R$ (see before (\\ref{SpectralRMixed})) and impose approximated radiation conditions on the artificial cuts. Denote by $n$ the unit normal to $\\partial\\Omega^R$ directed to the exterior of $\\Omega^R$, set \n\\[\n\\begin{array}{l}\n\\Gamma_1:=\\{R\\}\\times(-1\/2;1\/2)^2,\\quad \\Gamma_2:=(-1\/2;1\/2)\\times\\{R\\}\\times(-1\/2;1\/2),\\quad \\Gamma_3:=(-1\/2;1\/2)^2\\times\\{R\\}\\\\[4pt]\n\\Gamma_4:=\\{-R\\}\\times(-1\/2;1\/2)^2,\\ \\Gamma_5:=(-1\/2;1\/2)\\times\\{-R\\}\\times(-1\/2;1\/2),\\ \\Gamma_6:=(-1\/2;1\/2)^2\\times\\{-R\\}\n\\end{array}\n\\]\nand $\\Gamma:=\\cup_{j=1}^6\\Gamma_j$. On $\\Gamma_1$, according to (\\ref{decompo_vj}), we have\n\\[\n\\partial_n(v_1-\\psi_1w_{1}^{-})=\\partial_{z_1}(v_1-w_{1}^{-})=2^{-1\/2}r\\,U_{\\dagger}+\\dots\n\\]\nwhere the dots stand for terms which are small for large values of $R$. On the other hand on $\\Gamma_1$, there holds\n\\[\nv_1-w_{1}^{-}=2^{-1\/2}r\\,(R-i)\\,U_{\\dagger}+\\dots .\n\\]\nTherefore, this gives, still on $\\Gamma_1$, \n\\begin{equation}\\label{Approx_RC1}\n\\partial_n v_1=\\cfrac{v_1-w_{1}^{-}}{R-i}+\\partial_{z_1} w_{1}^{-}+\\dots=\\cfrac{v_1}{R-i}+w_{1}^{-}\\left(\\cfrac{1}{R+i}-\\cfrac{1}{R-i}\\right)+\\dots=\\cfrac{v_1}{R-i}-\\cfrac{2i}{R^2+1}\\,w_{1}^{-}+\\dots.\n\\end{equation}\nOn $\\Gamma_j$, $j\\ne1$, the situation is simpler because $v_1$ is outgoing in the corresponding branches and we have\n\\begin{equation}\\label{Approx_RC2}\n\\partial_n v_1=\\cfrac{v_1}{R-i}+\\dots .\n\\end{equation}\nFinally, using the Robin conditions (\\ref{Approx_RC1}), (\\ref{Approx_RC2}) as approximated radiation conditions, we consider the variational formulation\n\\begin{equation}\\label{Pb_Approximation}\n\\begin{array}{|l}\n\\mbox{Find }\\hat{v}_1\\in\\mrm{H}^1_{0}(\\Omega^R;\\partial\\Omega^R\\cap\\partial\\Omega)\\mbox{ such that for all }v\\in\\mrm{H}^1_{0}(\\Omega^R;\\partial\\Omega^R\\cap\\partial\\Omega) \\\\[6pt]\n\\displaystyle\\int_{\\Omega^R}\\nabla \\hat{v}_1\\cdot\\nabla v\\,dx-\\cfrac{1}{R-i}\\int_{\\Gamma} \\hat{v}_1 v\\,d\\sigma-(2\\pi)^2\\displaystyle\\int_{\\Omega^R}\\hat{v}_1 v\\,dx=-\\cfrac{2i}{R^2+1}\\int_{\\Gamma_1} w_{1}^{-} v\\,d\\sigma.\n\\end{array}\n\\end{equation}\nOne can prove that $\\hat{v}_1$ yields a good approximation of $v_1$ with an error which is exponentially decaying with $R$. In practice, we solve the problem (\\ref{Pb_Approximation}) with a P1 finite element method thanks to \\texttt{Freefem++}. Then replacing $v_1$ by $\\hat{v}_1$ in the exact formulas\n\\begin{equation}\\label{FormulaApproxCoef}\nr=\\cfrac{2}{R^2+1}\\,\\int_{\\Gamma_1} (v_1-w_{1}^{-})\\,w_{1}^{-}\\,d\\sigma,\\quad t=\\cfrac{2}{R^2+1}\\,\\int_{\\Gamma_4} v_1\\,w_4^{-}\\,d\\sigma,\\quad t^\\perp=\\cfrac{2}{R^2+1}\\,\\int_{\\Gamma_2} v_1\\,w_2^{-}\\,d\\sigma,\n\\end{equation}\nwe get an approximation of the threshold scattering matrix $\\mathbb{S}$ given by (\\ref{DefMatScaSimplifiee}). Finally with (\\ref{DefMatPol}), we obtain an approximation of the polarization matrix $\\mathbb{M}$ which appears in the $3\\times3$ spectral problems (\\ref{CompactForm}).\\\\\n\\newline\nOur computations give\n\\[\nr\\approx 0.66+0.11i,\\qquad t\\approx 0.08-0.70i,\\qquad t^{\\perp}\\approx -0.08-0.08i.\n\\]\nThe eigenvalues of $\\mathbb{S}\\in\\mathbb{C}^{6\\times6}$ are approximately equal to \n\\[\n0.44 - 0.9i\\mbox{ (simple)},\\quad 0.9 - 0.44i\\mbox{ (double)}\\quad\\mbox{ and }\\quad 0.58 + 0.81i\\mbox{ (triple).}\n\\]\nThey have modulus one which is consistent with the fact that $\\mathbb{S}$ is unitary. Moreover, we indeed observe that they are different from $-1$ which is coherent with the discussion following (\\ref{DefMatPol}) (absence of threshold resonance). On the other hand, for the coefficients of $\\mathbb{M}$ (see (\\ref{MatPolSimple})), we get\n\\begin{equation}\\label{ValuePolMat}\nr_m\\approx 0.08,\\qquad t_m\\approx -0.44,\\qquad t^{\\perp}_m\\approx -0.06.\n\\end{equation}\nWith these values, solving the $3\\times3$ eigenvalue problem (\\ref{CompactForm}) for $\\eta\\in[0;2\\pi)^3$, we find that the segment $\\aleph$ defined in (\\ref{defAleph}) satisfies\n\\begin{equation}\\label{NumAleph}\n\\aleph\\approx (-1.24;1.04).\n\\end{equation}\n\n\\noindent Let us mention that the Robin conditions (\\ref{Approx_RC1}), (\\ref{Approx_RC2}) are rather crude approximations of the exact radiation conditions. To get good errors estimates, we should take rather large values of $R$. However in practice, large $R$ are no so simple to handle and can create important numerical errors. Therefore a compromise must be found and we take $R=2.5$. Admittedly, this point should more investigated. \n\n\n\\section*{Apprendix}\n\n\\subsection{Friedrichs inequality}\nWe reproduce here the Lemma 5.1 of \\cite{BaMaNa17}.\n\\begin{lemma}\\label{LemmaPoincareFriedrichs}\nAssume that $a>0$. Then we have the Friedrichs inequality \n\\begin{equation}\\label{Fried_ineq}\n \\kappa(a)\\int_{0}^{1\/2}\\phi^2\\,dt \\le \\int_{0}^{+\\infty}(\\partial_t\\phi)^2\\,dt+a^2\\int_{1\/2}^{+\\infty}\\phi^2\\,dt,\\qquad \\forall \\phi\\in\\mrm{H}^1(0;+\\infty),\n\\end{equation}\nwhere $\\kappa(a)$ is the smallest positive root of the transcendental equation\n\\begin{equation}\\label{TransEqnLemma}\n\\sqrt{\\kappa}\\tan\\bigg(\\cfrac{\\sqrt{\\kappa}}{2}\\bigg)=a.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor $a>0$, consider the spectral problem \n\\begin{equation}\\label{Eigen1D}\n\\begin{array}{|rcll}\n-\\partial^2_t \\phi+a^2\\mathbbm{1}_{(1\/2;+\\infty)} \\phi &=& \\lambda(a)\\,\\mathbbm{1}_{(0;1\/2)}\\phi&\\mbox{ in }(0;+\\infty)\\\\[4pt]\n\\partial_t\\phi(0)&=&0\n\\end{array}\n\\end{equation}\nwhere $\\mathbbm{1}_{(1\/2;+\\infty)}$, $\\mathbbm{1}_{(0;1\/2)}$ stand for the indicator functions of the sets $(1\/2;+\\infty)$, $(0;1\/2)$ respectively. Let us equip $\\mrm{H}^1(0;+\\infty)$ with the inner product\n\\[\n(\\phi,\\phi')_a=\\int_{0}^{+\\infty}\\partial_t\\phi\\,\\partial_t\\phi'\\,dt+a^2\\phi\\,\\phi'\\,dt.\n\\]\nWith the Riesz representation theorem, define the linear and continuous operator $T:\\mrm{H}^1(0;+\\infty)\\to \\mrm{H}^1(0;+\\infty)$ such that\n\\[\n(T\\phi,\\phi')_a=\\int_{0}^{1\/2}\\phi\\,\\phi'\\,dt.\n\\]\nWith this definition, we find that $(\\lambda(a),\\phi)$ is an eigenpair of (\\ref{Eigen1D}) if and only if we have \n\\[\nT\\phi=(\\lambda(a)+a^2)^{-1}\\phi.\n\\]\nSince $T$ is bounded and symmetric, it is self-adjoint. Additionally the Rellich theorem ensures that $T$ is compact. This guarantees that the spectrum of (\\ref{Eigen1D}) coincides with a sequence of positive eigenvalues whose only accumulation point is $+\\infty$. Let us denote by $\\kappa(a)$ the smallest eigenvalue of (\\ref{Eigen1D}). From classical results concerning compact self-adjoint (see e.g. \\cite[Thm. 2.7.2]{BiSo87}), we know that\n\\begin{equation}\\label{Def_quotient}\n(\\kappa(a)+a^2)^{-1}=\\underset{\\phi\\in\\mrm{H}^1(0;+\\infty),\\,(\\phi,\\phi)_a=1}{\\sup} (T\\phi,\\phi)_a.\n\\end{equation}\nRearranging the terms, we find that (\\ref{Def_quotient}) provides the desired estimates (\\ref{Fried_ineq}). Now we compute $\\kappa(a)$. Solving the ordinary differential equation (\\ref{Eigen1D}) with $\\lambda(a)=\\kappa(a)$, we obtain, up to a multiplicative constant,\n\\[\n\\phi(t)=\\begin{array}{|ll}\n\\cos(\\sqrt{\\kappa(a)}t) & \\mbox{ for }t\\in(0;1\/2)\\\\[3pt]\nc\\,e^{-at} & \\mbox{ for }t\\ge 1\/2\n\\end{array}\n\\]\nwhere $c$ is a constant to determine. Writing the transmission conditions at $t=1\/2$, we find that a non zero solution exists if and only if $\\kappa(a)>0$ satisfies the relation (\\ref{TransEqnLemma}).\n\\end{proof}\n\\begin{lemma}\\label{LemmaPoincareFriedrichsR}\nAssume that $a>0$ and $R>1\/2$. Then we have the Friedrichs inequality \n\\begin{equation}\\label{Fried_ineq_R}\n \\kappa(a,R)\\int_{0}^{1\/2}\\phi^2\\,dt \\le \\int_{0}^{R}(\\partial_t\\phi)^2\\,dt+a^2\\int_{1\/2}^{R}\\phi^2\\,dt,\\qquad \\forall \\phi\\in\\mrm{H}^1(0;R),\n\\end{equation}\nwhere $\\kappa(a,R)$ is the smallest positive root of the transcendental equation\n\\begin{equation}\\label{TransEqnLemmaR}\n\\sqrt{\\kappa}\\tan\\bigg(\\cfrac{\\sqrt{\\kappa}}{2}\\bigg)=a\\,\\tanh(a(R-1\/2)).\n\\end{equation}\nTherefore, we have $\\lim_{R\\to+\\infty}\\kappa(a,R)=\\kappa(a)$ where $\\kappa(a)$ is the constant appearing in Lemma \\ref{LemmaPoincareFriedrichs}.\n\\end{lemma}\n\\begin{proof}\nThe demonstration is completely similar to the one of Lemma \\ref{LemmaPoincareFriedrichs} above. We find that the largest constant $ \\kappa(a,R)$ such that (\\ref{Fried_ineq_R}) holds coincides with the smallest eigenvalue of the problem\n\\[\n\\begin{array}{|rcll}\n-\\partial^2_t \\phi+a^2\\mathbbm{1}_{(1\/2;R)} \\phi &=& \\kappa(a,R)\\mathbbm{1}_{(0;1\/2)}\\phi&\\mbox{ in }(0;R)\\\\[4pt]\n\\partial_t\\phi(0)=\\partial_t\\phi(R)&=&0.\n\\end{array}\n\\]\nSolving it, we find that if $\\phi$ is a corresponding eigenfunction, up to a multiplicative constant, we have\n\\[\n\\phi(t)=\\begin{array}{|ll}\n\\cos(\\sqrt{\\kappa(a,R)}t) & \\mbox{ for }t\\in(0;1\/2)\\\\[3pt]\nc\\,\\cosh(a(t-R)) & \\mbox{ for }t\\in(1\/2;R)\n\\end{array}\n\\]\nfor some constant $c$. This times, writing the transmission conditions at $t=1\/2$, we find that a non zero solution exists when $\\kappa(a,R)$ satisfies (\\ref{TransEqnLemmaR}). Finally we obtain that $\\kappa(a,R)\\to\\kappa(a)$ because $\\tanh(a(R-1\/2))$ in (\\ref{TransEqnLemmaR}) tends to $1$ when $R\\to+\\infty$.\n\\end{proof}\n\n\\section{Acknowledgements}\nThe work of the second author was supported by the Russian Science Foundation, project 22-11-00046.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOver the past decade, advances in computing devices and navigation sensors accelerated the development and the range of missions USVs can undertake. USV are extensively utilised for applications such as seabed mapping, marine animals tracking, and maintenance of offshore facilities such as wind farms or oil pipelines. Compared to manned vehicles, a USV could be deployed for dangerous and boring tasks continuously in shallow and restricted waters \\cite{ma2018multi}. As a USV could carry heavy payload, it can install multiple energy resources to achieve long endurance performance \\cite{niu2018energy}. However, USV can only perceive dynamic sea condition in a short range. By contrast, UAV is capable of detecting large-scale sea condition at a high altitude and has better maneuverability than USV. However, UAV has limited flight endurance due to its limited battery capacity. By developing a system that can allow for a UAV to autonomously land on the USV, the flight range of UAVs will be considerably increased. UAV could also detect the USV surrounding environment at a high altitude \\cite{xiao2017uav}, which improves the detection range of USV. Moreover, UAV can be used as a communication relay between USV and ground station, which enlarges the communication range of USV. This research proposes an affordable system that allows the UAV and USV work collaboratively and autonomously for making uses of the strengths and diminishing the weaknesses of both UAVs and USVs.\n\n\n\\begin{figure} [!t]\n \\begin{center}\n \\includegraphics[width=3.4in]{UAVUSV.PNG}\n \\caption{3D printed UAV and USV.}\n \\label{fig1}\n \\end{center}\n\\end{figure}\n\n\n\\section{UAV and USV Design}\n\n\\subsection{UAV Design}\nTo protect the electronic components from sea water, we designed a waterproof UAV, which was 3D printed using Selective Laser Sintering (SLS) printer and PA2200 printing material, as shown in Fig.~\\ref{fig21}. The volume of the shell is 0.004006 cubic meters and its buoyancy can support a mass of $4.11kg$, which enables the UAV to float on water. To enable the UAV to localize the USV position precisely under various lighting conditions, we used an IR beacon detector on the bottom of UAV. The IR beacon detector was designed to filter most light noises and only detect the light within certain frequency spectrum emitted by the IR beacon, which was installed on the USV landing platform. The GPS, autopilot, battery and IR beacon detector were installed on the component tray, which is inside of the shell, as shown in Fig.~\\ref{fig21}.\n\n\n\\begin{figure}[h]\n \\begin{center}\n \\includegraphics[width=3.25in,height=1.6in]{MainstructualofUAV.PNG}\n \\caption{CAD design of waterproof UAV.}\n \\label{fig21}\n \\end{center}\n\\end{figure}\n\n\n\\subsection{Design of the USV and its Landing Platform}\n\nTo improve the stability of the landing platform, a catamaran USV was designed. The CAD design of hull and wooden keel are shown in Fig.~\\ref{fig24}. It is calculated to carry a payload of up to $32.5kg$ per hull when fully submerged. Polyethylene Terephthalate Glycol (PETG) filament was used to print hulls since it can produce strong parts with slight flexibility and has a relatively low shrinkage ratio compared to other filament types such as Acrylonitrile Butadiene Styrene (ABS) and Polylactic Acid (PLA). The final hull dimensions was $1.655m \\times 0.301m \\times 0.155m$ and the total mass was 8.1kg per hull. MotorGuide R3-30 trolling motors were selected and fitted via modified mounts that allowed the motors position adjustable, as shown in Fig.~\\ref{fig25}. The maximum thrust of motors can be up to $133.8N$. The weight of each hull with fully assembled motors and deck was $17.4kg$.\nAll control units of the USV, including raspberry pi, speed controller RoboClaw 2x30A, Battery, Telemetry Radio, Ublox Neo M8 compass and Emlid reach GPS, were installed in a waterproof box, as shown in Fig.~\\ref{fig26}, which was located in the centre of the landing platform with aluminium frame supports, as shown in Fig.~\\ref{fig27}.\n\n\\begin{figure}[hbt!]\n\\centering\n\\subfigure[]{\\label{fig24}\n\\includegraphics[width=0.23\\textwidth,height=0.16\\textwidth]{Woodenkeelandhull.PNG}}\n\\subfigure[]{\\label{fig25}\n\\includegraphics[width=0.23\\textwidth,height=0.16\\textwidth]{propulsion.jpg}}\n\\subfigure[]{\\label{fig26}\n\\includegraphics[width=0.23\\textwidth,height=0.16\\textwidth]{USVcontrolbox.PNG}}\n\\subfigure[]{\\label{fig27}\n\\includegraphics[width=0.23\\textwidth,height=0.16\\textwidth]{finalusvdesign.jpg}}\n\\caption{Catamaran USV: (a) Wooden keel and 3D printed hull (b) Propulsion system of USV (c) USV control box (d) Catamaran USV and landing platform}\n\\end{figure}\n\n\n\\section{Experiments and Discussion}\n\\subsection{Two-Phase Precise Landing Test}\nA two-phase landing method was proposed to land the UAV on the USV precisely using standard GPS and IR beacon. At first, the UAV navigates to the USV location using USV GPS data. Standard GPS data are not accurate enough to allow accurate landing on the platform due to the limited dimension. When the UAV is above the USV and localizes the IR beacon, precise landing using IR beacon will be activated. The accuracy of precise landing is evaluated using twenty tests. As shown in Fig.~\\ref{fig41}, the relative position between the IR beacon detector on the UAV and the IR beacon in the centre of the USV landing platform satisfies: $\\Delta X < 20cm$, $\\Delta Y < 20cm$. Note that the offset along y axis was caused by the wind during the tests. As the landing platform dimension is $1.2m \\times 1m$ that is larger than the landing derivation, the UAV could land on the USV safely. \n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\\label{fig41}\n\\includegraphics[width=3.0in]{Landingprecision.PNG}}\n\\subfigure[]{\\label{fig42}\n\\includegraphics[width=2.8in]{carrotchasing.PNG}}\n\n\\caption{UAV and USV experiments: (a) Accuracy results of twenty landing tests (b) Aerial view of USV control algorithm and path following algorithm test}\n\\end{figure}\n\n\\subsection{Control and Path Following Algorithm of USV}\nPI controller and PD controller were implemented for speed control and heading control \\cite{niu2016efficient}, respectively. To navigate the USV to follow the pre-defined path precisely, carrot chasing path following algorithm \\cite{niu2016efficient} was implemented on USV for generating required heading and speed command. As shown in Fig.~\\ref{fig42}, two waypoints were pre-defined by ground station. The USV was able to follow the pre-defined path, represented by the line between point1 and point2.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main results}\n\nThe geometric properties of random interfaces are a vast arena of study in rigorous statistical mechanics.\nTwo important classes of interface models are {\\em phase separation} models that idealize the boundary between a droplet of one substance suspended in another, and {\\em last passage percolation} models, where a directed path in independent local randomness maximizes a random weight determined by the environment.\n\nOne of the best known mathematical examples of phase separation is the two dimensional supercritical Ising model in a large box with negative boundary condition. Conditioned to have an atypically high number of positive spins in the box, the vertices in the plus phase tend to form a droplet surrounded by a sea of negative spins. The random phase boundary of such a droplet has been the object of intense study. \nWulff proposed that the profile of such constrained circuits would macroscopically resemble\na dilation of an isoperimetrically optimal curve. This was established rigorously in \\cite{DKS} and \\cite{IS}.\nVia the FK representation of the Ising model, \n one observes a similar situation in the setting of subcritical two dimensional percolation, where the analogous object is the boundary of the cluster containing the origin after this cluster is conditioned to be atypically large. The Wulff shape captures the macroscopic profile of the circuit in such models, but what of fluctuations? Several definitions may be considered that seek to capture fluctuation behaviour on the part of the circuit, including the deviation in the Hausdorff metric of the convex hull of the circuit from an appropriately scaled Wulff shape.\nAlternative definitions, more local in nature, serve better to capture the transition in circuit geometry from local Brownian randomness to a smoother profile dictated by the constraints of global curvature.\nIn fact, a pair of definitions is natural, one to capture the longitudinal distance at which the transition takes place, and the second to treat the orthogonal inward deviation of the interface at this transition scale. \nTo specify the characteristic longitudinal distance, we may note that the convex hull of the circuit is a polygonal path that is composed of planar line segments or {\\em facets}; we may treat the typical or maximum facet length as a barometer of the transition from the shorter scale of local randomness to the longer scale of curvature. Latitudinally, we may note that any point in the circuit has a {\\em local roughness}, given by its distance from the convex hull boundary. The typical or maximum local roughness along the circuit is a latitudinal counterpart to facet length. \n Alexander \\cite{Alexander}, and Hammond \\cite{AH1,AH2,AH3} analysed such conditioned circuit models and determined that when the area contained in the circuit is of order $n^2$, so that the circuit has diameter of order $n$, facet length and local roughness scale as $n^{2\/3}$ and $n^{1\/3}$. A similar situation is witnessed when a parabola $x \\to t^{-1}x^2$ is subtracted from a two-sided Brownian motion $B:\\mathbb{R} \\to \\mathbb{R}$. When $t > 0$ is large, facets of the motion's convex hull have length $\\Theta(t^{2\/3})$ and inward deviation $\\Theta(t^{1\/3})$.\nThis phenomenon is expected to be universal when the local structure of the interface is Brownian and other examples include \na Brownian bridge pinned at $-T$ and $T$ and conditioned to remain above the semi-circle of radius $T$ centred at the origin which was analysed by Ferrari and Spohn \\cite{FS}. \n\nThe second class of random geometric paths we have mentioned are last passage percolation models. These models form part of a huge, Kardar-Parisi-Zhang, class of statistical mechanical models in which a path through randomness is selected to be extremal for a natural weight determined by that randomness. The maximizing paths are often called {\\em polymers}. \nThe fluctuation behaviour of a length $n$ polymer with given endpoints may be gauged either in terms of the scale of deviation of its weight from the mean value, or by the scale of deviation of say the polymer's midpoint from the planar line segment that interpolates the polymer's endpoints. The two deviations are given by scales $n^{1\/3}$ and $n^{2\/3}$. \nThese were first proved for planar Poissonian\ndirected last passage percolation, in the seminal work of Baik, Deift and Johansson \\cite{BDJ99}. Since then, this and other integrable models in the same KPZ universality class have been extensively analysed and detailed information about this model, both geometric and algebraic, has been obtained \\cite{J00, LM01, LMS02}. \n\nIt is of much interest to study models that combine phase separation and path minimization (or maximization) in random environment. Consider an environment with local independent randomness, and the circuit through that randomness whose weight determined by that randomness is extremal among those circuits that trap a given area. Studying the random geometry of such a circuit is a problem of extremal isoperimetry. Taking the randomness to be supercritical percolation, Itai Benjamini conjectured the first-order, Wulff-like behaviour of the boundary of the set in the supercritical percolation cluster attaining the so called anchored expansion constant. This was proved by~\\cite{BLPR} by showing that the curve in the limit solves a natural isoperimetric variational problem; this has recently been extended to higher dimensions by Gold \\cite{G16}. Given the role of facet length and local roughness in capturing the local random to global curvature transition in circuit geometry, the natural next problem is to understand the scaling exponents of such objects, which is the pursuit we undertake in this paper. \nOur choice of model preserves the qualitative features of the problem of interface fluctuation in an isoperimetrically extremal droplet in supercritical percolation and at the same time ensures that key algebraic aspects of KPZ theory can be harnessed to yield sharp fluctuation estimates. \n\n\nThe particular setting we consider in this paper is planar Poissonian \ndirected last passage percolation. \nTo impose the required properties, we study this model under a quadratic curvature constraint: that is, we force the best path to move away from the straight line and have a quadratic curvature on the average. This is done naturally by considering the longest upright path in a Poissonian environment joining $(0,0)$ and $(n,n)$ which has the additional \\textit{area trap} property that the area under the curve is at least $(\\frac{1}{2}+\\alpha)n^2$ for some $\\alpha\\in (0,\\frac{1}{2})$. Note that from the discussion above it follows that the unconstrained longest path stays close to the diagonal and hence encloses area $(\\frac{1}{2}+o(1))n^2$. \nPostponing precise statements until later (see Section \\ref{scale-exp}), our main results quantify the competition between the global parabolic nature and local behaviour guided by KPZ relations of the contour.\nThey show that the maximum facet length of the contour's least concave majorant scales as $n^{3\/4+o(1)}$, while the inward deviation from the concave majorant (local roughness) \nscales as $n^{1\/2+o(1)}$.\n\nWe also establish a law of large numbers for the length of the optimal path. The proof proceeds by setting up a variational problem, as is natural in such contexts (see \\cite{DZ2}, \\cite{BLPR}). Perhaps surprisingly, however, the problem turns out to have a very explicit solution. The geometric information about the limit shape of the constrained polymer is also used as input in some of the arguments about fluctuations. \nThe question of fluctuation in the context of isoperimetrically extremal circuits in supercritical percolation can be formulated as a first passage percolation analogue of our setting of last passage percolation. Although the first passage percolation model is not exactly solvable, it, too, is believed to be in the KPZ universality class, and the geodesics there are believed to have the same $n^{2\/3}$ scaling of transversal fluctuation as in our case \n(see \\cite{FPPsurvey} and the references therein). \nThus our results suggest that a certain universality is being witnessed and, according to this belief, we formulate a conjecture concerning percolation. \nWe elaborate on this and a number of other interesting questions in Section \\ref{oq}.\n\nFinally, we discuss briefly the key inputs used in our paper and how it contrasts with the other examples of fluctuation results already mentioned.\nThe results in \\cite{AH1,AH2,AH3} crucially used the refined understanding and geometric estimates for percolation clusters while a study of area trapping planar Brownian loop \\cite{HP} used well known estimates for Brownian motion. Exact expressions involving Brownian motion conditioned to stay over a parabola was also the key ingredient in the proofs in \\cite{FS}. \nHowever, in our setting, even though the unconstrained model has integrable properties, the lack of general geometric understanding as one deviates slightly from integrable models causes a big challenge for us. \nNonetheless, using certain known facts about the unconstrained model and their robust variants established recently in \\cite{BSS14} as blackbox estimates, we rigorously establish local roughness exponents for our model which we henceforth call the \\emph{Area Trapping Polymer} model (see next section for precise definitions). \nThus, this work is an example where one can use inputs from the integrable literature to make geometric conclusions about settings which are beyond the exactly solvable world (see also \\cite{BSS14}).\nWe believe a general program of developing such geometric arguments would lead to more robust proofs which could work for settings that are non-integrable but are variants of solvable models. \n\n\\subsection{Model Definitions}\\label{def1012}\nWe now recall the planar Poissonian directed last passage percolation model. \n\nLet $\\Pi$ be a homogeneous rate one Poisson Point Process (PPP) on the plane. A partial order on $\\mathbb{R}^2$ is given by $(x_1,y_1)\\preceq (x_2,y_2)$ if $x_1\\leq x_2$ and $y_1\\leq y_2$. \nFor $u\\preceq v$, a directed path $\\gamma$ from $u$ to $v$ is a piecewise linear path that joins points $u=\\gamma_{0}\\preceq \\gamma_1 \\preceq \\cdots \\preceq \\gamma_{k}= v$ where each $\\gamma_i$ for $i\\in [k-1]$ (throughout the article we will adopt the standard notation $[n]=\\{1,2,\\ldots,n\\}$)\nis a point of $\\Pi$. \nDefine the length of $\\gamma$, denoted $|\\gamma|$, to be the number of $\\Pi$-points on $\\gamma$. \n\\begin{definition}\\label{geodesic1} \n Define the last passage time from $u$ to $v$, denoted by $L(u,v)$, to be the maximum of $|\\gamma|$ as $\\gamma$ varies over all directed paths from $u$ to $v$. There may be several maximizing paths between $u$ and $v$, and throughout the paper we will refer to the top most path (it is easy to see that the top most path is well defined here) among those, as the \\emph{geodesic} between $u$ and $v$ and denote it by $\\gamma(u,v)$. \n \\end{definition}\n\nWe will often call $\\gamma(u,v)$ as the polymer between $u$ and $v$ and $|\\gamma(u,v)|$ as the polymer length\/weight respectively.\nNext we introduce a constraint in this classical model. \n\n\\subsubsection{Area Trapped by a path}\\label{def10}\nConsider a path $\\gamma$ between the origin $(0,0)$ and a point $(x,y)$ in the positive quadrant. The area trapped by the path $\\gamma$, denoted by $A(\\gamma)$, is defined to be the area of the closed polygon determined by the $x$-axis, the vertical line segment joining $(x,0)$ to $(x,y)$ together with the line segments of the path $\\gamma$. Let $\\gamma_n$ denote the geodesic between $(0,0)$ and $(n,n)$. Well-known facts about Poissonian LPP readily imply that $A(\\gamma_{n})= (\\frac{1}{2}+o(1))n^2$ asymptotically almost surely. We constrain the model and consider maximizing paths subject to trapping a much larger area. To this end fix $\\alpha\\in (0,\\frac{1}{2})$, and let \n\\begin{equation}\n\\label{e:cop}\nL_{\\alpha}(n):=\\max\\left\\{|\\gamma|: \\gamma~\\text{path from}~(0,0)~\\text{to}~(n,n)~\\text{and}~ A(\\gamma)\\geq \\big(\\tfrac{1}{2}+\\alpha\\big)n^2\\right\\}.\n\\end{equation}\nIt is easily seen that, among the paths that attain this maximum, there is almost surely exactly one that traps the least area. This path will be called $\\Gamma_{\\alpha,n}$. We will write $\\Gamma_{n}$ provided that the context clarifies the value of $\\alpha$ in question. We shall call $\\Gamma_n$ the constrained (or $\\alpha$-constrained) geodesic.\nIn analogy with the phase separation example in percolation mentioned the introduction, it might seem more natural to consider the family of down right paths joining $(n,0)$ and $(0,n)$ since all of these curves enclose the origin. However because of the obvious underlying symmetry, and to take advantage of standard notational conventions, throughout the sequel we will consider the contour joining the origin to the point $(n,n)$.\n\n\nOur main objects of interest are two quantities that measure local regularity of the constrained geodesic $\\Gamma_n$. The following definitions are illustrated by Figure \\ref{def}.\n\n\\begin{definition}\n\\label{d:mlrmfl}\nLet ${\\rm conv}(\\Gamma_{\\alpha, n})$ denote the convex hull of the polygon determined by the path $\\Gamma_{\\alpha, n}$ and the coordinate axes. Let $\\Gamma^{*}_{\\alpha, n}$ denote the closure of the polygonal part of the boundary of ${\\rm conv}(\\Gamma_n)$ between $(0,0)$ and $(n,n)$ above the $x$-axis. Thus $\\Gamma^{*}_{\\alpha, n}$ is the least concave majorant of $\\Gamma_{\\alpha,n}$ and is an union of finitely many line segments. These segments will be called {\\rm facets}.\nDefine {\\rm{maximum facet length}} of $\\Gamma_n$, denoted $\\mathrm{MFL}(\\Gamma_{\\alpha, n})$ to be the maximum Euclidean length of the facets. For $x\\in \\Gamma_{\\alpha, n}$, let ${\\rm d}(x,\\Gamma^{*}_{\\alpha, n})$ denote the distance from $x$ to $\\Gamma^*_{\\alpha, n}$. This is a natural notion of the \\emph{local roughness at $x$}. \nDefine the {\\rm{maximum local roughness}} of $\\Gamma_n$, \ndenoted $\\mathrm{MLR}(\\Gamma_n)$ by\n$$\\mathrm{MLR}(\\Gamma_{\\alpha, n}):= \\sup\\{x\\in \\Gamma_{\\alpha, n}: {\\rm d}(x,\\Gamma^{*}_{\\alpha, n})\\}.$$\n\\end{definition} \n\nIn the following subsections we present our main results. We first state our results regarding the fluctuation exponents of $\\Gamma_{\\alpha, n}$.\n\\subsection{Scaling exponents for geodesic geometry}\\label{scale-exp}\nThe sense in which we capture the exponents is stronger and easier to state for the lower bounds, and so we begin with them. \n\n\\begin{theorem}\n\\label{t:mfllb}\nFix $\\alpha \\in (0,\\frac{1}{2})$ and $\\varepsilon>0$. Then there exists $c=c(\\alpha,\\varepsilon)>0$ such that for all large enough $n,$\n$$\\mathbb{P}(\\mathrm{MFL}(\\Gamma_n) \\geq n^{3\/4-\\varepsilon})\\ge 1-e^{-n^c}.$$ \n\\end{theorem}\n\nAs we have mentioned, the scaling exponent for transversal fluctuation of point-to-point geodesic in unconstrained Poissonian last passage percolation is known to be $2\/3$ \nThis fact, put together with the above theorem yields the following lower bound on maximum local roughness.\n\n\\begin{theorem}\n\\label{t:mlrlb}\nFix $\\alpha \\in (0,\\frac{1}{2})$ and $\\varepsilon>0$. Then there exists $c=c(\\alpha, \\varepsilon)>0$ such that for all large enough $n,$\n$$\\mathbb{P}(\\mathrm{MLR}(\\Gamma_n) \\geq n^{1\/2-\\varepsilon})\\ge 1-e^{-n^c}.$$ \n\\end{theorem}\n\nRegarding the matching upper bound, \nwe prove that, with high probability, there exists a dense set of $\\alpha \\in (0,2^{-1})$ for which the maximum length of the facets away from the boundary is bounded above by $n^{3\/4+o(1)}$. To make this precise, fix $\\delta \\in (0,\\pi\/4)$, and consider a facet in $\\Gamma_{\\alpha, n}$ with endpoints $A$ and $B$\nrecorded in clockwise order. Setting $O = (n,0)$, let $\\theta_{A}$ denote the acute angle that $OA$ makes with the $y$-axis and $\\theta_{B}$, the acute angle that $OB$ makes with the $x$-axis; see Figure~\\ref{def}. \n\n\\begin{definition}\n\\label{interior1}\nThe facet $AB$ is called $\\delta$-interior if $\\min (\\theta_{A}, \\theta_{B})\\geq \\delta$.\n\\end{definition} \n\nNote that the union of the $\\delta$-interior facets forms a polygonal path.\n(The union could be empty, but we will infer later from Theorem~\\ref{lln2}\nthat this event has an exponentially small probability.)\n\\footnote{Note that at this point, it is not a priori clear if the set of $\\delta-$interior facets is non-empty. However, this will be a consequence of Theorem \\ref{lln2}, stated later, which implies for any $\\delta$, the length of all the facets \nwill be less than $O(\\delta n)$ with exponentially small failure probability.} Let $A_0$ and $B_0$ denote the extremities of this union path, and let $\\Gamma_{\\delta, \\alpha, n}$ denote the subpath of $\\Gamma_{\\alpha,n}$ between $A_0$ and $B_0$. Define the maximum $\\delta$-interior facet length of $\\Gamma_{\\alpha, n}$, denoted by $\\mathrm{MFL}(\\Gamma_{\\delta, \\alpha,n})$, to be the maximum length of the $\\delta$-interior facets. Define the $\\delta$-interior maximum local roughness, denoted by $\\mathrm{MLR}(\\Gamma_{\\delta, \\alpha, n})$ by altering Definition \\ref{d:mlrmfl} so that now the supremum is taken over all $x \\in \\Gamma_{\\delta, \\alpha, n}$. \n\\begin{definition}\\label{good10}\nFix $\\varepsilon>0$. We say $\\alpha\\in (0,\\frac{1}{2})$ is $(n,\\varepsilon,\\delta)$-good if $\\mathrm{MFL}(\\Gamma_{\\delta, \\alpha, n}) \\leq n^{3\/4+\\varepsilon}$. \n\\end{definition}\nHere is our upper bound concerning the maximum length of facets.\n\n\\begin{theorem}\n\\label{t:mflub}\nFix $\\varepsilon>0$, $\\delta\\in (0,\\pi\/4)$ and an interval $[\\alpha_1, \\alpha_2] \\subset (0,\\frac{1}{2})$. Let $I_{n, \\alpha_1,\\alpha_2}$ denote the set of $(n,\\varepsilon, \\delta)$-good $\\alpha \\in [\\alpha_1, \\alpha_2]$. Then there exists $c=c(\\varepsilon,\\delta,\\alpha_1,\\alpha_2)>0$ such that the probability that $I_{n, \\alpha_1,\\alpha_2} \\not= \\emptyset$ is at least $1-e^{-n^c}$ for all large enough $n.$\n\\end{theorem} \n\nOur final principal result concerning exponents asserts that, for $\\alpha\\in I_{n,\\alpha_1,\\alpha_2}$, it is highly likely that $\\mathrm{MLR}(\\Gamma_{\\delta, \\alpha, n})\\geq n^{1\/2+\\varepsilon}$. Thus we obtain, with high probability, an $n^{1\/2+\\varepsilon}$ upper bound for interior maximum local roughness for a dense, though possibly $n$-dependent and random, set of~$\\alpha$. \n\n\\begin{theorem}\n\\label{t:mlrub}\nIn the setting of Theorem \\ref{t:mflub}, there exists $c>0$, such that the event that there exists $\\alpha\\in I_{n, \\alpha_1,\\alpha_2}$ such that \n$\\mathrm{MLR}(\\Gamma_{\\delta, \\alpha, n})> n^{1\/2+\\varepsilon},$ occurs with probability at most $e^{-n^c} $ for all large enough $n$.\n\\end{theorem}\n\n\\begin{figure}[h] \n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.38\\textwidth]{flatfacet.pdf} &\\quad\\includegraphics[width=0.4\\textwidth]{fluccurv.pdf} \\\\\n(a) & (b)\n\\end{tabular}\n\\caption{ (a) The three different curves correspond to $\\psi_{\\alpha,n},\\Gamma_{\\alpha,n}, \\Gamma^*_{\\alpha,n}$. The dashed lines indicate the $\\delta-$ interior and hence the facets inside the sector bounded by them, are the $\\delta-$interior facets.\n(b)The scale at which the competition from two sources are equal to each other.}\n\\label{def}\n\\end{figure}\n\nWe now state our results about law of large numbers for $L_{\\alpha}(n)$ and the constrained geodesic. \nAlthough the route taken here of setting up an appropriate variational problem is by now classical \\cite{DZ2}, we point out that it was not at all obvious that the solution can be explicitly described. Moreover some of the consequences of the results in the following section are used as geometric inputs for the fluctuation results stated before. \n\\subsection{Law of Large Numbers}\nFor the unconstrained model, a straightforward subadditivity argument yields that $\\frac{\\mathbb{E} L_{n}}{n}$ converges to a limit. The evaluation of the limiting constant is classical: \\cite{VerKer77, LogShep77} showed that the limit equals $2$ by using Young tableaux combinatorics and the RSK correspondence (see also \\cite{AD95}). However, for the constrained model, the subadditive structure is lost and it is not clear a priori that a law of large numbers for $L_{\\alpha}(n)$ exists. Our first result here is to establish the law of large numbers for the area trapping polymer model; we are also able to evaluate the limiting constant implicitly as a function of $\\alpha$. \n\nGiven $\\alpha \\in (0,1\/2)$, let $c_{\\alpha}$ be given implicitly by the following equation:\n\\begin{equation}\\label{implicit1}\n\\frac{1+c_{\\alpha}}{c_{\\alpha}}\\left(1-\\frac{\\log(1+c_{\\alpha})}{c_{\\alpha}}\\right)=\\frac{1}{2}+\\alpha.\n\\end{equation}\nOne can see that the function $f(c)=\\frac{1+c}{c}[1-\\frac{\\log(1+c)}{c}]$ is strictly increasing \\footnote{$f'(c)=\\frac{(2+c)\\log(1+c)-2c}{c^3}$ and $\\frac{d}{dc}[(2+c)\\log(1+c)-2c]=\\log(1+c)-\\frac{c}{1+c} >0$.} in $c$, and converges to $1\/2$ and $1$ at $0$ and $\\infty$ respectively. \nLet ${\\mathrm{w}}_{\\alpha}= \\sqrt{1+c_{\\alpha}}\\frac{\\log(1+c_{\\alpha})}{c_{\\alpha}}$.\n\\begin{theorem} \n\\label{t:lln} \nFor any $\\alpha\\in (0,1\/2)$ \n$$\\mathbb{E} L_{\\alpha}(n)=2{\\mathrm{w}}_{\\alpha} n+o(n)$$\nas $n\\to \\infty$.\n\\end{theorem}\n\nNotice that ${\\mathrm{w}}_{\\alpha}\\to 1$ as $\\alpha\\to 0$; hence the above theorem \nis consistent with the result in the classical unconstrained case. In the unconstrained model, one also has a law of large numbers for the geodesic, i.e., it is known that the geodesic is concentrated around the straight line joining $(0,0)$ and $(n,n)$. More precisely, under the rescaling that takes the $n\\times n$ square to a unit square, the geodesic converges almost surely in Hausdorff distance to the diagonal of the unit square \\cite{DZ2} (as mentioned before, the precise order of the transversal fluctuations are known to be $n^{2\/3}$ as a consequence of integrability). Even though we do not have any exactly solvable structure in the constrained model, we can establish a similar law of large numbers here too asserting that the constrained geodesic concentrates around a deterministic curve. Moreover, we can identify the limiting curve in a fairly explicit manner. \n\nLet $\\psi_{\\alpha}:[0,1]\\to [0,1]$ be defined as \n$\\psi_{\\alpha}(x)=\\frac{(1+c_{\\alpha})x}{1+c_{\\alpha}x}$ where $c_{\\alpha}$ is as above. Also let $\\psi_{\\alpha,n}: [0,n]\\to [0,n]$ be the $n$ blow up of $\\psi_{\\alpha}$ i.e. $\\psi_{\\alpha,n}(x)=n \\psi_{\\alpha}(x\/n)$. We denote by ${\\rm dist}(\\cdot,\\cdot)$ the Hausdorff distance. \nThe following theorem is our law of large numbers for the constrained geodesic.\n\n\\begin{theorem}\\label{lln2} For any $\\alpha \\in (0,1\/2)$ and $\\Delta>0$, there exists $c=c(\\alpha,\\Delta)$ such that, for all large enough $n$, it is with probability at least $1-e^{-cn}$ that $${\\rm dist}(\\Gamma_{\\alpha,n},\\psi_{\\alpha,n})\\le \\Delta n.$$\n\\end{theorem}\n\nIn particular, this theorem states that to first order, the constrained geodesics behave like a given smooth curve. This result will be useful to us while studying the scaling exponents for local roughness of the constrained geodesic, in particular, when we see to rule out long and flat facets (see Theorem \\ref{flat0}).\n\n\\subsection{Open questions and future directions}\\label{oq}\n\n\nBelow we list below several interesting questions for future research:\n\\begin{enumerate}\n\\item The upper bound Theorem \\ref{t:mflub} is weaker than the lower bound Theorem \\ref{t:mfllb}. Strengthening the former is a natural open problem. \n\\item By definition, $A(\\Gamma_{\\alpha,n})\\ge (\\frac{1}{2}+\\alpha)n^2$. The typical order of $A(\\Gamma_{\\alpha,n})- (\\frac{1}{2}+\\alpha)n^2$ remains unknown. \n\n\\item For $\\alpha \\in (0,1\/2)$,\nwhat is the typical deviation of $\\Gamma_{\\alpha,n}$ from the curve $\\psi_{\\alpha,n}$? What is the order of fluctuations of $L_{\\alpha}(n)$?\n\\item In a phase separation problem, \\cite{AH1,AH2,AH3} determines the polylogarithmic corrections to both elements of the $(2\/3,1\/3)$ (facet length,local roughness) exponent pair. The powers of the logarithm are $(1\/3,2\/3)$. Finding such corrections for the \\emph{Area Trapping Polymer} model would refine the identification of the exponent pair $(3\/4,1\/2)$ made in this article and suggested by the first point.\nThe first two points will be discussed further in \nSection~\\ref{areafluc}.\n\\end{enumerate}\n\\noindent\n\\textbf{Supercritical Percolation}: We end this section with a conjecture regarding fluctuation exponents in the context of supercritical percolation on the nearest neighbor graph on $\\mathbb{Z}^2$. As already mentioned before, \\cite{BLPR} settles a conjecture of Benjamini regarding the limit shape of isoperimetrically extremal sets. Formally, for supercritical percolation, with the origin $\\mathbf{0}$ conditioned to be in the infinite cluster $\\mathcal{C}_{\\infty}({\\bf{0}})$, the authors in \\cite{BLPR} consider the `anchored isoperimetric profile', i.e. for any $r>0$ they look at the set $\\mathcal{B}_{r}$ which solves the following isoperimetric problem:\n$$\\inf \\left\\{\\frac{|\\partial B|}{|B|}: {\\bf{0}}\\in B, B\\subset \\mathcal{C}_{\\infty}({\\bf{0}}) \\text{ is connected}, |B|\\le r \\right\\}$$ where $\\partial B$ denotes the edge boundary of $B,$ restricted to $\\mathcal{C}_{\\infty}(\\bf{0})$ (for more details see \\cite{BLPR}).\nThe main result in \\cite{BLPR} shows the convergence of the set $\\mathcal{B}_r$ as $r\\to \\infty$ after suitable renormalization, to a deterministic Wulff crystal, in the Hausdorff sense. To go beyond first order behaviour one has to understand local geometric properties of the boundary of $\\mathcal{B}_r$. \nThe extremal circuit broadly has to satisfy the:\n\\begin{enumerate}\n\\item Volume condition\n\\item Extremal isoperimetry condition\n\\end{enumerate}\nThe heuristic now is that the former is a global constraint while the latter is only local. \nNamely, each local part of the boundary does not feel the volume constraint $|\\mathcal{B}_r|\\le r,$ and thereby just tries to move through the `open' edges while trying to minimize the number of open edges that it cuts across, since these are precisely the edges that contribute to $|\\partial \\mathcal{B}_r|$.\n This brings us within the realm of first passage percolation predicted to be in the KPZ universality class as well(see \\cite{FPPsurvey} for more on first passage percolation).\n Thus Theorems, \\ref{t:mfllb}, \\ref{t:mlrlb}, \\ref{t:mflub} and \\ref{t:mlrub} can be thought of as rigorous counterparts to the above discussion in the integrable last passage percolation setting and combined with the above discussion allows us to leads us to conjecture fluctuation exponents for the above model. Below we formulate a precise statement in a slightly simpler setting: \n \n Recall the standard definition of the dual graph of the nearest neighbor lattice on $\\mathbb{Z}^2$. \nGiven a supercritical bond percolation environment on the edge set of $\\mathbb{Z}^2$ with density $p>p_c(\\mathbb{Z}^2)=1\/2$, for every positive integer $n,$ consider the set $\\mathcal{D}_{n^2},$ of dual circuits (simple loop consisting of dual edges) enclosing a connected subset of $\\mathbb{Z}^2$ of size $n^2$ and containing the origin such that the number of primal open edges in the percolation environment that the circuit cuts through is minimized. Note that such a circuit is not necessarily unique due to the discrete nature of the problem. \nThus the above model is an exact analogue of the model considered in this paper, in the context of first passage percolation in a bond percolation environment.\nWe now state precisely our conjecture: \n\n\\noindent\n\\textbf{Conjecture}:\nConsider bond percolation on the nearest neighbor lattice on $\\mathbb{Z}^2$ with any supercritical parameter value \n $p>p_c(\\mathbb{Z}^2)$. The random variables \n$$\\max_{D\\in \\mathcal{D}_{n^2}}\\left|\\frac{\\log(\\mathrm{MFL}(D))}{\\log n}-3\/4\\right| \\, \\, \\text{ and } \\, \\, \\max_{D\\in \\mathcal{D}_{n^2}}\\left|\\frac{\\log(\\mathrm{MLR}({D}))}{\\log n}-1\/2\\right|$$ converge to zero in probability as $n$ grows to infinity\nwhere for any $D\\in \\mathcal{D}_{n^2}$, the maximum facet length $\\mathrm{MFL}(D)$ and maximum local roughness $\\mathrm{MLR}(D)$ are defined in the same way as in this paper by considering the convex hull of the points in $D$.\n\nWe end with the remark that the problem considered in \\cite{BLPR}\ncorresponds to a similar first passage percolation problem where the environment has bounded dependence range and hence should have the same fluctuation behaviour as the simpler model\njust described. \n \n\\subsection{Organization of the rest of the article}\n In Section \\ref{s:aux2} we collect some preliminary probabilistic results: some for the constrained model and a few from the unconstrained model. The results for the unconstrained model follow from the sharp moderate deviation estimates in \\cite{LM01, LMS02} and the consequences established in \\cite{BSS14}. In Section~\\ref{s:var}, we set up and solve the variational problem for the law of large numbers and the following Section \\ref{s:lln} is devoted to the proofs of Theorem \\ref{t:lln} and Theorem \\ref{lln2}: the law of large numbers for the length of constrained geodesic and the path itself.\n We next turn to the proofs of the results on the scaling exponents. In Section \\ref{s:lb} we provide proofs of Theorems \\ref{t:mfllb} and Theorem \\ref{t:mlrlb}. The proofs of Theorem \\ref{t:mflub} and Theorem \\ref{t:mlrub} are completed in Section \\ref{s:ub}. Proofs of some of the auxiliary results stated and used throughout the article are postponed to Section \\ref{es}. \n\n\n\\subsection*{Acknowledgments}\nThe authors thank Marek Biskup, Craig Evans and Ofer Zeitouni for useful discussions. S.G.'s research is supported by a Miller Research Fellowship at UC Berkeley. A.H. is supported by NSF grant DMS-1512908. \n\n\\section{Important probability estimates}\n\\label{s:aux2}\nIn this section we gather the probabilistic inputs needed for our proof. First we shall record a few useful facts about the length and geometry of the constrained geodesics that will be used repeatedly later. The bulk of this section will then recall results about the unconstrained model. These results are all consequences of the exactly solvable nature of the Poissonian LPP model and can be derived starting with the basic integrable ingredients of the exactly solvable model obtained by Lowe, Merkl and Rolles \\cite{LM01,LMS02}. Some of these consequences were established and used recently in \\cite{BSS14} and we quote the relevant results here.\n\nWe start with some basic facts about the constrained model.\n\n\\subsection{Some Basic Results for the Area Trapping Polymer Model}\nWe first start with concentration for $L_{\\alpha}(n)$. In absence of integrability we use the standard Poinc\\'are inequality techniques and obtain a concentration at $n^{1\/2+o(1)}$ scale. \n\n\\begin{theorem}\\label{concentration1} Fix any $\\alpha \\in (0,1\/2)$. Then there exists a constant $C>0$ such that for all $t>0$ $$\\mathbb{P}(|L_{\\alpha}(n)-\\mathbb{E}(L_{\\alpha}(n))| \\ge t) \\le Ce^{-t^{2}\/Cn \\log^{2}(n)}.$$\n\\end{theorem} \nThe proof is standard and is postponed until the end of the paper in Section \\ref{es}. \n\nOur next result rules out flat facets.\nThis is a consequence of Theorem \\ref{lln2}, which asserts that it is extremely unlikely that any interior facets (which are those in the bulk) have very shallow or steep gradient.\n\\begin{theorem}\\label{flat0} For any small enough $\\delta>0$ there exists $\\gamma>0$ such that, with probability $1-e^{-cn}$, all $\\delta-$interior facets make an angle with the $x-$axis which lies in the interval $(\\omega,\\pi\/2-\\omega)$.\n\\end{theorem}\n\n \\begin{proof} \nWe start by recalling a simple fact: \n for two facets of $\\Gamma^*_{\\alpha,n}$ with starting points $u_1,u_3$ and ending points $u_2, u_4$ respectively where $u_1 \\preceq u_2 \\preceq u_3 \\preceq u_4,$ by convexity of $\\Gamma^*_{\\alpha,n}$, the angle made with the $x-$axis by the facet $(u_1,u_2)$ is larger than the angle made by $(u_3,u_4)$. \n\n\nConsider any $\\delta-$interior facet $(u_1,u_2),$ and let $u_{1}=(x,y)$.\nAlso let $L_1,L_2$ be the straight lines joining the origin to $u_1$ and $(n,n)$ to $u_2$ respectively. \nLet $v_1$ and $v_2$ be the points of intersection of $L_1$ and $L_2$ with $\\psi_{\\alpha,n}$. Theorem \\ref{lln2} implies that, for any $\\varepsilon > 0$, with probability at least $1-e^{-cn},$ the bound \n$$\\max (|u_1-v_1|,|u_2-v_2|)\\le \\varepsilon n$$\nholds for all $\\delta$-interior facets, simultaneously. \nNow, by the convexity of $\\Gamma^{*}_{\\alpha, n}$, the gradient of the facet $(u_1,u_2)$ is between the gradient of the lines joining $(0,0)$ to $u_1$ and $(n,n)$ to $u_2$. Choosing $\\varepsilon$ to be much smaller than $\\delta,$ we see that the gradient of the facet $(u_1,u_2)$ plus an error of $O(\\varepsilon)$ lies between the gradients of the lines joining the origin and $v_1$ and $(n,n)$ and $v_2$ respectively. \nThus we are done by choosing $\\varepsilon$ to be small enough and using the strict convexity of $\\psi_{\\alpha,n}$; (see Figure \\ref{def}(a) for illustration). \n\\end{proof}\n\nWe will later need Theorem \\ref{flat1}, a strengthening \nof the above result, which is uniform \nin $\\alpha$. \nWe next state useful results that concern the unconstrained, exactly solvable, model. All of these are corollaries of the following moderate deviation estimates.\n\n\\subsection{Moderate deviation estimates}\nRecall the polymer length \n$L(\\cdot,\\cdot)$ from Definition \\ref{geodesic1}.\nLet $u=(u_1,u_2)\\preceq v=(v_1,v_2)$ be such that $|u_1-v_1||u_2-v_2|=t$; i.e., $t$ is the area of the rectangle whose opposite corners are $u$ and $v$. Notice that, by scale invariance of the Poisson point process, the distribution of $L(u,v)$ is a function merely of $t$.\n\n\\begin{theorem}[\\cite{LM01,LMS02}]\n\\label{t:moddev}\nFix $\\kappa>1$. Let $u$, $v$ as above be points such that the straight line joining $u$ and $v$ has gradient $m$, where \n$m\\in ( \\kappa^{-1} , \\kappa)$. There exist positive constants $s_0$, $t_0$, $C$ and $c$ depending only on $\\kappa$ such that, for all $t>t_0$ and $s>s_0$,\n$$\\mathbb{P}[|L(u,v)-2\\sqrt{t}| \\geq st^{1\/6}]\\leq Ce^{-cs^{3\/2}}.$$\n\\end{theorem}\n\nObserve that the above theorem implies that $|\\mathbb{E} L(u,v) - 2\\sqrt{t}|=O(t^{1\/6})$; and hence similar tail bounds are true for the quantity $|L(u,v)-\\mathbb{E} L(u,v)|$. When we make use of Theorem \\ref{t:moddev}, it will often be in order to obtain tail bounds for the quantities $|L(u,v)-\\mathbb{E} L(u,v)|$. Also the results of \\cite{BDJ99} establish that $t^{-1\/6} (L(u,v)-2\\sqrt{t})$ converges weakly to the GUE Tracy-Widom distribution (which is defined for example in \\cite{BDJ99}). This law has negative mean, leading to the next bound. \n\n\\begin{equation}\n\\label{e:mean}\n\\mathbb{E} L(u,v)\\leq 2\\sqrt{t}-Ct^{1\/6}\n\\end{equation}\nfor some $C>0$.\n\n\n\\subsection{Transversal fluctuations of point-to-point geodesics}\nLet $u=(u_1,u_2)\\preceq v=(v_1,v_2)$. Let $y=p_{u,v}(x)$ denote the equation of the line segment joining $u$ and $v$.\nFor any path $\\gamma$ from $u$ to~$v$, define the maximum transversal fluctuation of the path $\\gamma$ by \n$${\\rm TF}(\\gamma):= \\sup_{x\\in [u_1,v_1]} \\{\\sup |p_{u,v}(x)-y|: (x,y)\\in \\gamma\\}.$$\nFor points $u=(u_1,u_2)$ and $v=(v_1,v_2)$, we call $|v_1-u_1|$ the $x$-coordinate distance between $u$ and $v$ and similarly define the $y$-coordinate distance. \n\nTransversal fluctuations for paths between $(0,0)$ and $(n,n)$ were shown to be $n^{2\/3+o(1)}$ with high probability in \\cite{J00}. The following more precise estimate was established in \\cite{BSS14}.\n\n\\begin{theorem}[Tails for Transversal Fluctuations, \\cite{BSS14}]\n\\label{t:tftail}\nFix $\\kappa>1$. Let $u\\preceq v$ be points such that the $x$-coordinate distance between $u$ and $v$ is $r$ and the gradient of the line joining $u$ and~$v$ is $m$, where $m\\in (\\kappa^{-1}, \\kappa)$. Then, for the geodesic $\\Gamma$ from $u$ to $v$, \n$$\\mathbb{P}[{\\rm TF}(\\Gamma) > sr^{2\/3}]\\leq Ce^{-cs}$$\nfor $\\kappa$-dependent constants $C,c$, and $r,s$ sufficiently large. \n\\end{theorem}\n\nThe proof of Theorem \\ref{t:tftail} \nfrom \\cite{BSS14} in fact shows something more. Any path that has transversal fluctuations much bigger than $r^{2\/3}$\nis not only unlikely to be a geodesic; it is typically much shorter than a geodesic. In particular, the proof of Theorem 11.1 in \\cite{BSS14} implies the next result (which can also be derived more straightforwardly).\n\n\\begin{theorem}[Paths with off-scale ${\\rm TF}$ are uncompetitive]\n\\label{t:tftail1}\nFix $\\kappa>1$ and $\\varepsilon>0$. Let $u\\preceq v$ be points such that the $x$-coordinate distance between $u$ and $v$ is $r$ and the gradient of the straight line joining $u$ and $v$ is $m$ where $m\\in ( \\kappa^{-1}, \\kappa)$. \nLet $\\mathcal{E}=\\mathcal{E}_{u,v}$ denote the event that there exists a path $\\gamma$ from $u$ to $v$ with ${\\rm TF}(\\gamma) \\geq r^{2\/3+\\varepsilon}$ and $|\\gamma|> \\mathbb{E}[L(u,v)]-r^{1\/3+\\varepsilon\/10}$ (we suppress the dependence of $\\varepsilon$ in $\\mathcal{E}$ for brevity). Then there exists $c=c(\\varepsilon)>0$ such that\n$$\\mathbb{P}(\\mathcal{E}) \\leq e^{-r^{c}}$$ \nfor $r$ sufficiently large. \n\\end{theorem}\n\nThe stretched exponential nature of these estimates allow us to prove uniform properties of the noise space which will be convenient for some of the arguments later. This point is a theme in this article: points to which we seek apply the above results may be special, so that estimates that are uniform over all points will be needed. \nFor the next result, recall that our noise space is nothing other than a rate one Poisson point process in the box $[0,n]^2$. \nFor $u,v\\in [0,n]^2$, let $\\mathds{L}(u,v)$ denote the line joining $u$ and $v$. Now, for any $u\\preceq v$, the pair $(u,v)$ is called $\\kappa$-steep if the gradient \nof $\\mathds{L}(u,v)$ lies in the interval $[\\kappa^{-1},\\kappa]$. Lastly,\nfor $\\tau>0$, \n\\begin{equation}\\label{notation540}\nS(u,v)= S_{\\tau}(u,v):=\\left\\{ \\begin{array}{cc}\n|u-v|^{1\/3+\\varepsilon\/10} & \\text{ if } |u-v|>n^{\\tau},\\\\ \nn^{2\\tau\/3} & \\text{ otherwise }.\n\\end{array}\\right. \n\\end{equation}\n\n\\begin{corollary}\n\\label{uniform1}\nFix $\\varepsilon>0$, $\\kappa>1$, and $\\tau$ sufficiently small. Then there exists $c=c(\\varepsilon,\\tau)$ such that\n$$\\mathbb{P}\\left(\\bigcup_{u,v: (u,v) \\text{is } \\kappa-\\text{steep}}\n|L(u,v)-\\mathbb{E}(L(u,v))|\\ge S(u,v)\\right) \\le e^{-n^c}.$$\n\\end{corollary}\n\\begin{proof}\nThis follows by using Theorem \\ref{t:moddev} and a standard coarse graining argument. Since we are not attempting to prove optimal bounds, we simply partition the box into squares of area one. Thus the corner points are the lattice points, i.e., elements of the set $\\{0,1,\\ldots,n\\}\\times\\{0,1,\\ldots,n\\}$. The proof now follows by first proving the desired statement for lattice points $u,v$. This just follows by Theorem \\ref{t:moddev} and a union bound over all pairs of lattice points $u,v$ (of which there are $n^2$). The proof is then completed by showing that, since any box of area one is unlikely (with stretched exponentially small failure probability) to contain more than $n^{\\varepsilon\/100}$ points, for any $u$ and $v$, one can approximate $L(u,v)$ by $L(u_*,v_*)$, where $u_*,v_*$ are the nearest lattice points.\n\\end{proof}\n\nWe now specify a modification of the event $\\mathcal{E}_{u,v}$ from Theorem \\ref{t:tftail1}. Let $\\mathcal{E}^{*}_{u,v}$ denote the event $\\mathcal{E}_{u,v}$ where the permitted error term $r^{1\/3+\\varepsilon\/10}$ is instead taken to be the quantity~$S(u,v)$ defined a few moments ago. The next corollary is proved along the same lines as Theorem~\\ref{t:tftail1}, in combination with the arguments in the proof of Corollary \\ref{uniform1},\nand we omit the proof. \n\\begin{corollary}\n\\label{uniform0}\nFix $\\varepsilon>0$, $\\kappa>1$, and $\\tau$ small enough. Then there exists $c=c(\\varepsilon,\\tau)$ such that\n$$\\mathbb{P}\\left(\\bigcup_{u,v:|(u,v) \\text{is } \\kappa-\\text{steep}}\\mathcal{E}^*_{u,v}\\right) \\le e^{-n^c}.$$\n\\end{corollary}\n\n\\subsection{Paths between points in an on scale parallelogram}\nNext we control the deviation of lengths between pairs of points within certain parallelograms. We first need some definitions. Let $U(r,m,h)$ denote a parallelogram with the properties that:\n\\begin{enumerate}\n\\item[(a)] One pair of parallel sides are vertical.\n\\item[(b)] The other pair of parallel sides have gradient $m$.\n\\item[(c)] The (horizontal) distance between the vertical sides is $r$.\n\\item[(d)] The height of the vertical sides is $h$.\n\\end{enumerate}\nNote that this defines a unique parallelogram up to translation. This will be enough for our purposes because the underlying noise field is translation invariant. \n\n\\begin{theorem}[\\cite{BSS14}]\n\\label{t:par1}\nLet $m\\in (\\kappa^{-1},\\kappa)$ and $h=W r^{2\/3}$ for some $W>0$. Then there exist $(\\kappa,W)$-dependent constants $C,c>0$, such that, for all sufficiently large $r$ and $s$, \n$$ \\mathbb{P}\\left( \\sup_{u\\in A, v\\in B} (L(u,v)- \\mathbb{E} L(u,v)) \\geq sr^{1\/3} \\right) \\leq Ce^{-cs},$$\nand\n$$ \\mathbb{P}\\left( \\inf_{u\\in A, v\\in B} (L(u,v)- \\mathbb{E} L(u,v)) \\leq -sr^{1\/3} \\right) \\leq Ce^{-cs},$$\nwhere $A$ and $B$ respectively denote the right third and the left third of $U(r,m,h)$.\n\\end{theorem}\n\nThe next theorem states that, even if the paths are restricted to stay inside the parallelogram, the fluctuation remains on scale. \nLet $L(u,v;U)$ denote the length of the longest path from $u$ to $v$ that does not exit $U$. \n\n\\begin{theorem}[\\cite{BSS14}]\n\\label{t:par2}\nUnder the assumptions of Theorem \\ref{t:par1}, there exist $(\\kappa,W)$-dependent constants $C,c>0$ such that, for all sufficiently large $r$ and $s$, \n$$ \\mathbb{P}\\left( \\inf_{u\\in A, v\\in B} (L(u,v;U)- \\mathbb{E} L(u,v)) \\leq -sr^{1\/3} \\right) \\leq Ce^{-cs^{1\/2}}$$\nwhere $A$ and $B$ respectively denote the right third and the left third of $U(r,m,h)$.\n\\end{theorem}\n\n\\subsection{Paths constrained in a thin parallelogram}\nThe next set of results shows that, if a path is constrained to have much smaller than typical transversal fluctuation, then it must have much smaller length than a typical geodesic's.\n\\begin{theorem}[Paths with small ${\\rm TF}$ are uncompetitive]\n\\label{t:constrained}\nFix $\\varepsilon>0$. Consider the parallelogram $U=U(r,m,h)$, where $m\\in (\\kappa^{-1},\\kappa)$ and $h=r^{2\/3-\\varepsilon}$. Let $u_0$ and $v_0$ denote the midpoints of the vertical sides $A_0$ and $B_0$ of $U$. Then, for some $c=c(\\varepsilon)>0$,\n$$\\mathbb{P}\\left(L(u_0,v_0; U) \\geq \\mathbb{E} L(u_0,v_0)- r^{1\/3+\\varepsilon\/3}\\right) \\leq e^{-r^{c}}$$\nfor $r$ sufficiently large.\n\\end{theorem}\n\nThe proof of Theorem \\ref{t:constrained} follows a strategy, by now well known, that has been used to show Gaussian fluctuation of paths constrained to stay in a thin rectangle in \\cite{CD13} in the context of first passage percolation. In the context of LPP, the Gaussian fluctuation has recently been shown in \\cite{DPM17}, and tail bounds are proved in a more general context in the preprint \\cite{BH17}. We now provide a sketch of Theorem~\\ref{t:constrained}'s proof. \n\n\\begin{proof}[Sketch of Proof]\nWithout loss of generality let us assume $m=1$. Also, we can replace the parallelogram $U$ by the rectangle $U',$ one of whose pairs of sides is parallel to the line segment $u_0v_0$, with the other pair having midpoints $u_0$ and $v_0$ \nand length $r^{2\/3-\\varepsilon}$. Divide the rectangle $U'$ into $K$ equal parts (each of width $\\frac{r}{K}$) by parallel line segments perpendicular to $u_0v_0$. Let $L_{i}$ denote the left side of the $i$-th rectangle and let $u_{i}$ be its midpoint. Now let $\\gamma_{i}$ be the best \npath (i.e. with maximal value of $|\\gamma|$) between $L_{i}$ and $L_{i+1}$ that stays within $U'$. It follows from Theorem \\ref{t:par1} that it is extremely likely that $|\\gamma_{i}|-L(u_i,u_{i+1})\\ll (K^{-1}r)^{1\/3}$. It follows that up to an error of order much smaller than $K^{2\/3}r^{1\/3}$ we can approximate $L(u_0,v_0; U')$ by $\\sum_{i} L(u_i,u_{i+1})$. Now observe that $L(u_i,u_{i+1})$ are independent and identically distributed with mean $2r\/K-C(K^{-1}r)^{1\/3}$ (here we use~\\eqref{e:mean}) and standard deviation of the order of $ (K^{-1}r)^{1\/3}$. By standard concentration inequalities, one then shows that $\\sum_{i} L(u_i,u_{i+1})$ concentrates around $2r-K^{2\/3}r^{1\/3}$ at scale $K^{1\/6}r^{1\/3}$. By choosing $K\\gg r^{\\varepsilon\/2}$ properly one gets the result. \n\\end{proof}\n\nIn the same vein, we have the following corollary.\n\\begin{corollary}\n\\label{uniform} \nFix $\\varepsilon>0$, $\\kappa>1$, and $\\tau$ small enough. Then there exists $c=c(\\varepsilon,\\tau)$ such that, with probability at least $1-e^{-n^c}$,\n$$\n\\displaystyle{\\sup_{u,v:|u-v|\\ge n^{\\tau}, (u,v) \\text{is } \\kappa-\\text{steep}} L(u,v,U)-\\mathbb{E}(L(u,v)) \\le -|u-v|^{\\frac{1}{3}+\\varepsilon}}.\n$$\n\\end{corollary} \n\n\n\\subsection{Estimates for One-Sided Geodesics}\nWe end this section by considering geodesics in a different constrained model. Baik and Rains \\cite{br01,br} considered increasing\npaths from $(0,0)$ to $(n,n)$ that lie above the diagonal line joining the points. Recall that $L(u,v)$ denotes the length of the longest increasing path between points $u \\preceq v$. Let $L^{\\boxslash}(u,v)$ denote the length of the longest path between $u$ and $v$ restricted to lie \nabove the line joining $u$ and $v$ and, similarly to Definition \\ref{geodesic1}, let $\\gamma^{\\boxslash}(u,v)$ denote the corresponding uppermost one-sided geodesic between $u$ and $v$. Moreover, let $L^{\\boxslash}(n)$ denote $L^{\\boxslash}(u,v)$ in the special case that $u=(0,0)$ and $v=(n,n)$. Baik and Rains \\cite{br01,br} proved that $\\mathbb{E} L^{\\boxslash}(n)=2n+o(n)$ and that the fluctuations are again of order $n^{1\/3}$ (although in this case the scaling limit is different; it is the GSE Tracy Widom distribution instead of the GUE Tracy Widom distribution). We shall need the corresponding moderate deviation estimates, consequences of Theorem \\ref{t:par2}. \n\n\\begin{theorem}\n\\label{baikrains} \nLet $u,v$ be as in the hypothesis of Theorem \\ref{t:moddev}. There exist positive constants $s_0, t_0, C$ and $c$ such that, for all $t>t_0$ and $s>s_0$,\n$$\\mathbb{P}[|L^{\\boxslash}(u,v)-2\\sqrt{t}| \\geq st^{1\/6}]\\leq Ce^{-cs^{1\/2}}.$$\n\\end{theorem}\n\n\\begin{proof}\nThe upper tail result follows from Theorem \\ref{t:moddev} due to $L^{\\boxslash}(u,v)\\leq L(u,v)$. For the lower tail, we use Theorem \\ref{t:par2} with the straight line joining $u$ and $v$ being the bottom side of the parallelogram and use the fact that $\\mathbb{E} L(u,v)=2\\sqrt{t}- \\Theta(t^{1\/6})$.\n\\end{proof}\n\nWe need a uniform version of this result. Recall $S(u,v)=S_{\\tau}(u,v)$ from \\eqref{notation540}.\n\\begin{corollary}\n\\label{uniform2}\nFix $\\varepsilon>0$, $\\kappa>1$, and $\\tau > 0$ small enough. Then there exists $c=c(\\varepsilon,\\tau)$ such that, with probability at least $1-e^{-n^c}$,\n$$\n\\displaystyle{\\sup_{u,v:(u,v) \\text{is } \\eta-\\text{steep}} L^{\\boxslash}(u,v)-\\mathbb{E}(L(u,v)) \\ge -S(u,v)} .\n$$\n\\end{corollary}\n\nProofs of Corollaries \\ref{uniform0} \\ref{uniform}, \\ref{uniform2} follow from the corresponding theorems for fixed points, in the same way as Corollary \\ref{uniform1} follows from Theorem \\ref{t:moddev}. We omit the details. \n\n\\section{A Variational Approach to the Constrained Geodesic}\n\\label{s:var}\nWe now move towards proving Theorem \\ref{t:lln} and Theorem \\ref{lln2}.\nDeuschel and Zeitouni \\cite{DZ1,DZ2} studied in detail the hydrodynamic limit of the unconstrained geodesic for inhomogeneous point processes. We follow their strategy broadly and study the limit of the constrained curve by means of appropriate variational problems. \nWe first explain the idea. Fix $\\alpha\\in (0,\\frac{1}{2})$ for the rest of this section. First let us assume that a limiting continuous curve $\\phi$ of the constrained geodesics exists after scaling. This curve $\\phi:[0,1] \\to [0,1]$ will be continuous, non-decreasing and surjective. By the area constraint, $\\int_0^1\\phi(s)ds \\ge \\tfrac{1}{2}+\\alpha$. Now heuristically, since in practice there is always a little bit of area excess, one can approximate $\\phi$ by a piecewise affine function at a scale local enough that each piece is exempt from the area constraint; and we can then use the law of large numbers for the unconstrained geodesic (see Theorem \\ref{t:moddev}) at every local scale and sum over them. This argument suggests that the approximating path will have length $2J({\\phi})n +o(n)$ where \n$$J({\\phi})=\\int_{0}^1\\sqrt {\\dot{\\phi}(s)}ds.$$\nIt thus seems that we should maximize $J({\\phi})$ among all curves $\\phi$ satisfying the area constraint, and the maximum will be the constant appearing in the required law of large numbers. We now proceed to make this precise. \nLet $\\mathcal{B}$ be the collection of all right-continuous non-decreasing functions from $[0,1]$ to $[0,1]$. \nThus $\\mathcal{B}$ is in bijection with the set of all sub-probability measures on $[0,1]$. Now, for any $\\phi\\in \\mathcal{B},$ by the Lebesgue decomposition theorem we can write \n \\begin{equation}\\label{decomposition23}\n \\phi=\\phi_{ac}+\\phi_{s}\n \\end{equation}\n in a unique way as a sum of a pair of sub-probability measures, with $\\phi_{ac}$ being the absolutely continuous part and $\\phi_s$ the singular part (with respect to Lebesgue measure). \nNote that this implies $\\phi_{ac}$ has a derivative \n$\\dot \\phi_{ac}$ almost everywhere \\footnote{Throughout the article, for any function $f$ on $[0,1],$ which is differentiable almost everywhere, we will denote its derivative by $\\dot f$. Recall, that this does not necessarily imply $\\int_{0}^x \\dot f(s)ds=f(x)$ for all $x$.} \nsuch that, for any $0\\le x\\le 1$, we have $$\\int_{0}^x \\dot \\phi_{ac}(s)ds =\\phi_{ac}(x)$$ while $\\phi_s$ is almost surely flat and hence has derivative $0$ almost surely. Thus $\\dot \\phi=\\dot \\phi_{ac}$ almost surely.\nAlso define \n\\begin{equation}\\label{variational1}\nJ(\\phi)= \\int_{0}^1\\sqrt {\\dot{\\phi}(s)}{\\rm d}s .\n\\end{equation} \nLet \n\\begin{equation}\\label{restr304}\n\\mathcal{B}_{\\alpha}=\\left\\{\\phi\\in \\mathcal{B}: \\int_0^1\\phi(s)ds \\ge \\tfrac{1}{2}+\\alpha\\right\\}.\n\\end{equation}\nWe shall often omit the subscript $\\alpha$. Finally let \n\\begin{equation}\\label{functional}\nJ_{\\alpha}=\\sup_{\\phi \\in \\mathcal{B}_{\\alpha}} J(\\phi).\n\\end{equation}\n\nThe first step is to show existence and uniqueness of the solution. \n\n\\begin{proposition}\n\\label{opt1} \nThere exists a unique element $\\psi=\\psi_{\\alpha} \\in \\mathcal{B}$ that attains the supremum in \\eqref{functional}.\n\\end{proposition}\n\nNote that the function $\\psi=\\psi_{\\alpha}$ in Proposition \\ref{opt1} will be the same in the statement of Theorem~\\ref{lln2}. The existence and uniqueness parts of Proposition \\ref{opt1} have separate proofs. We first state the existence result.\n\n\\begin{lemma}\n\\label{l:exist}\nThere exists $\\psi\\in \\mathcal{B}$ that achieves the supremum in \\eqref{functional}.\n\\end{lemma}\n\nThe proof of Lemma \\ref{l:exist} is technical: it uses a compactness argument on the space of probability measures following a similar argument from \\cite{DZ2}. We postpone the proof until Section \\ref{es}. \n\nIn the next part we show uniqueness. \n\n\\begin{lemma}\n\\label{l:uni}\nSuppose $\\psi_1$ and $\\psi_2$ are in $\\mathcal{B}_{\\alpha}$ and satisfy $J(\\psi_i)= J_{\\alpha}$ for $i=1,2$. Then $\\psi_1=\\psi_2$. \n\\end{lemma}\n\nThe proof of Lemma \\ref{l:uni} is rather straightforward once we establish the following technical lemma that rules out the possibility that any of the optimizing functions have a nontrivial singular part. \n\n\\begin{lemma}\n\\label{l:ac}\n Let $\\psi \\in \\mathcal{B}_{\\alpha}$ be such that $J(\\psi)= J_{\\alpha}$. Then $\\psi$ corresponds to a probability measure which is absolutely continuous with respect to Lebesgue measure.\n\\end{lemma}\n\nHere is the basic idea of the proof of this lemma. From the results of \\cite{DZ2} it follows that without the area constraint the optimizing curve is the diagonal line, i.e., the preferred gradient for the graph of the function is one. If the singular part is non-trivial, then the graph of the function $\\psi$ may be expected to have a flat piece. Because the gradient of the graph has to be one on average due to boundary conditions, it follows that there must be a piece of the graph having gradient away from one. We shall show that one can modify the flat and the steep parts of the curve locally by pieces with more moderate gradient in a way that increases not only the value of the functional $J$; and thereby optimality will be contradicted. Also, clearly in the case that $\\psi$ does not have total mass one, one can add to it another function to make it (or, more precisely of course, to bring it into correspondence with) a probability measure, while also increasing the value of $J(\\psi),$ and thus contradict maximality in this case also. Thus $\\psi$ is also a probability measure. The details of the proof are postponed to Section \\ref{es}.\n\n\n\nWe can now prove the uniqueness result using Lemma \\ref{l:ac}. \n\n\\begin{proof}[Proof of Lemma \\ref{l:uni}]\nWe use the fact that the square-root function is strictly concave. Given any $\\psi_1$ and $\\psi_2$ as in the statement of the lemma, we consider $\\psi=\\frac{\\psi_1+\\psi_2}{2}$. Clearly, $\\psi$ satisfies the area constraint and $\\dot{(\\frac{\\psi_1+\\psi_2}{2})}=\\frac{\\dot {\\psi_1}+\\dot{\\psi_2}}{2}$. Thus using Jensen's inequality $J(\\psi)$ is necessarily larger than $J(\\psi_1)=J(\\psi_2)$ unless $\\dot \\psi_1=\\dot \\psi _2$ almost surely. Since by the previous result the absolutely continuous parts contain mass one, $\\psi_1(0)=\\psi_2(0)=0$. Hence for all $0\\le x\\le1 $ \n$$\\psi_1(x)=\\int_0^{x}\\dot \\psi_1(s)ds=\\int_0^{x}\\dot \\psi_2(s)ds=\\psi_2(x).$$ Thus we are done. \n\\end{proof}\n\nIt still remains to identify the optimizer $\\psi$ in \\eqref{functional}; in particular we need to show this is the same $\\psi$ as defined in Theorem \\ref{lln2}. To this end, we shall now record some properties of the unique optimizer $\\psi$ that will be useful later. We show that the derivative is almost surely positive and decreasing. The proofs are easy and will be postponed until Section \\ref{es}.\n\\begin{lemma}\\label{decreasing1} \nLet $\\psi=\\psi_{\\alpha}$ denote the unique optimizer in Proposition \\ref{opt1}. Then\n\\begin{enumerate}\n\\item[(i)] The derivative $\\dot \\psi$ is almost surely decreasing i.e. there is a set of full measure on which it is decreasing. \n\\item[(ii)] $\\dot \\psi$ is almost surely positive.\n\\end{enumerate}\n\\end{lemma}\n\n\n\\subsection{Identifying the curve $\\psi_{\\alpha}$}\nIn this subsection we determine the curve $\\psi_{\\alpha}$ and show that it is the same as the curve in Theorem \\ref{lln2}. We first start with the following proposition.\n\n\\begin{proposition}\n\\label{solution1} \nGiven $\\alpha \\in (0, 1\/2)$ there exists $c=c_\\alpha>0,$ such that\n$\\psi_\\alpha(x)=\\frac{(1+c)x}{1+cx}$.\n\\end{proposition}\n\n\\begin{proof}The proof proceeds by showing that, given any $0 \\alpha$ implies $c_{*}J(\\psi)$. This contradicts the optimality of $\\psi$, and completes the proof. \n\\end{proof}\nObserve that, $J_{\\alpha}=\\sqrt{1+c_{\\alpha}}\\frac{\\log(1+c_{\\alpha})}{c_{\\alpha}}$.\nThis expression is the same as the constant $\\mathrm{w}_{\\alpha}$ in Theorem \\ref{t:lln} which is consistent with our heuristic explanation at the beginning of this section. We prove this theorem next. \n\n\\section{Law of large numbers}\n\\label{s:lln}\nWith the preparation from the previous section we now turn to the proof of Theorem \\ref{t:lln}. Fix $\\alpha\\in (0,\\frac{1}{2})$ as before. The idea of the proof is as follows. We first show that if a path follows approximately the blow up $\\psi_{\\alpha, n}$ of the deterministic curve $\\psi$ (and also satisfies the area constraint) then it is overwhelmingly likely that the path has length at least $2(\\mathrm{w}_{\\alpha}-\\varepsilon)n$ for $\\varepsilon$ arbitrarily small. We further show that any path satisfying the area constraint is extremely likely to have length less than $2(\\mathrm{w}_{\\alpha}+\\varepsilon)n$. We begin with the following trivial but useful lemma that is an immediate consequence of continuity and monotonicity of ${\\rm{w}}_{\\alpha}$.\n\n\n\\begin{lemma} Given any $\\alpha \\in (0,\\frac{1}{2}),$ for all small enough $\\varepsilon>0,$ there exist $\\delta_1,\\delta_2 > 0$ such that $|{\\rm{w}}_{\\alpha}-{\\rm{w}}_{\\alpha_1}| < \\varepsilon$ if $|\\alpha-\\alpha_1|\\le \\delta_1 $ and $|{\\rm{w}}_{\\alpha}-{\\rm{w}}_{\\alpha_1}| > \\varepsilon$ if $|\\alpha-\\alpha_1| \\ge \\delta_2$.\n\\end{lemma}\nNext we need a couple of preparatory lemmas about appropriate discretisations. Let $I_{\\delta}=\\{0,\\delta,2\\delta,\\ldots,1\\}$ be the discretisation of the unit interval; (we will take $\\delta$ to be the reciprocal of a positive integer to avoid rounding errors). Consider any non-decreasing function $L: [0,1] \\to [0,1]$ that corresponds to an absolutely continuous measure on $[0,1]$: by identifying $L$ with its graph, we shall interpret $L$ as an increasing path on the unit square directed from $(0,0)$ to $(1,1)$. \nAlso let $L_{\\delta}$ be the piecewise affine function that agrees on $I_\\delta$ with $L$.\n\n\\begin{lemma} \n\\label{l:approx3}\nWe have $\\left | \\int_{0}^1[L(x)-L_{\\delta}(x)]{\\rm{d}}x \\right | \\le 2\\delta$.\n\\end{lemma}\n\n\\begin{proof}First notice that by integration by parts or by Fubini's theorem, for any absolutely continuous function $f$ on $[0,1],$ $\\int_0^1 f(x){\\rm{d}}x =\\int_0^1 \\dot f(x) (1-x){\\rm{d}}x $. The lemma now follows from the observation that $\\left |\\int_{0}^1 x \\dot L(x){\\rm{d}}x-\\int_{0}^1 x \\dot L_{\\delta}(x){\\rm{d}}x\\right|\\le 2\\delta$.\n{Let $x_{i}$ be the midpoint of the interval $I_\\delta^{(i)}=[i\\delta,(i+1)\\delta]$. \nNotice that by definition $\\dot L_{\\delta}$ is constant on $I_\\delta^{(i)}$ and is equal to $\\frac{1}{\\delta}\\int_{I_\\delta^{(i)}}\\dot L(x){\\rm{d}}x $. \nThus \n$$\n\\sum_{i=0}^{\\frac{1}{\\delta}-1}\\int_{I_{\\delta}^{(i)}} x_i \\dot L(x) {\\rm{d}}x=\\sum_{i=0}^{\\frac{1}{\\delta}-1}x_i \\int_{I_{\\delta}^{(i)}} \\dot L(x){\\rm{d}}x=\\sum_{i=0}^{\\frac{1}{\\delta}-1}x_i \\int_{I_{\\delta}^{(i)}} \\dot L^{\\delta}(x){\\rm{d}}x \n=\\sum_{i=0}^{\\frac{1}{\\delta}-1}\\int_{I_{\\delta}^{(i)}} x \\dot L^{\\delta}(x)=\\int_0^1 x \\dot L^{\\delta}(x)\n$$\nThe proof now follows by observing $|\\int_{0}^1 x \\dot L(x){\\rm{d}}x -\\int_{0}^1 x \\dot L^{\\delta}(x){\\rm{d}}x | \\le \\sum_{i=0}^{\\frac{1}{\\delta}-1}\\int_{I_{\\delta}^{(i)}} |x - x_i| \\dot L(x){\\rm{d}}x\\le \\delta$.\n}\n\\end{proof}\nWe now discretise the vertical direction as well. \nLet us choose $\\eta\\ll \\delta$ (to be specified exactly later), and let $B_{\\delta,\\eta}= I_{\\delta} \\times I_{\\eta}$. Given $L$ as before let us now set, \n$L_{\\delta,\\eta}$ to be the piecewise linear curve determined by the points $(i\\delta, \\eta \\lfloor \\frac{L(i\\delta)}{\\eta} \\rfloor)$.\nClearly for all $x,$ $|\\dot L_{\\delta}-\\dot L_{\\delta,\\eta}| \\le \\frac{2\\eta}{\\delta},$ and hence for $\\eta\\le \\delta^2,$\n\\begin{equation}\\label{area14}\n\\bigg\\vert \\, \\int_{0}^1[L(x)-L_{\\delta,\\eta}(x)]{\\rm{d}}x \\, \\bigg\\vert \\le 4\\delta. \\end{equation}\n\n\n\nBefore proceeding further we make a few comments about notation. Recall that till now, our underlying noise space has been a point process $\\Pi$ on $\\mathbb{R}^2$ and we have been concerned about geodesics in $[0,n]^2$. However in the last section while solving variational problems we switched to a normalized picture where every path lives in $[0,1]^2$. We will continue with this convention throughout this section. Equivalently the noise space can be thought of as a Poisson point process of intensity $n^{2}$ in $[0,1]^2$. In an abuse of notation, we will define $L(u,v)$ for two points $u,v \\in [0,1]^2$ to be the length of the geodesic $\\gamma(u,v)$ between $u$ and $v$ in the same noise space. In light of properties of Poisson process under scaling, \n{Moreover, all the estimates regarding $\\gamma(u,v)$ stated in the previous sections continue to hold after the appropriate variable change. We omit further elaboration.} \nNext, we state a uniform version of Theorem \\ref{t:moddev} in the large deviation regime. \n \n \n \n\\begin{lemma}\n\\label{uni100} Fix $\\delta,\\eta$ as above.\nThere exists $c=c(\\delta,\\eta)$ such that, simultaneously for all $x\\in I_{\\delta}$ and $y_2> y_1 \\in I_{\\eta}$ for all large $n$:\n\\begin{enumerate}\n\\item With probability at least $1-e^{-cn},$ $$L((x,y_1),((x+\\delta),y_2))- 2 n \\sqrt{\\delta(y_2-y_1)}\\le \\eta \\delta n,$$ \n\\item With probability at least $1-e^{-cn^2},$ $$L((x,y_1),((x+\\delta),y_2))- 2 n \\sqrt{\\delta(y_2-y_1)}\\ge - \\eta \\delta n.$$ \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} The proof follows from the large deviation probabilities for the length of unconstrained geodesics in \\cite{sepLDP}.\nIn particular the theorem states that for any $x,y_1,y_2$ as in the statement of the theorem, \n\\begin{align*}\n\\mathbb{P}(L((x,y_1),((x+\\delta),y_2))- 2 n \\sqrt{\\delta(y_2-y_1)}\\ge \\eta \\delta n)&\\le e^{-cn},\\\\\n\\mathbb{P}(L((x,y_1),((x+\\delta),y_2))- 2 n \\sqrt{\\delta(y_2-y_1)} -\\eta \\delta n)&\\le e^{-cn^2},\n\\end{align*}\nfor some constant $c=c(\\delta,\\eta)$.\nA union bound over points $x,y_1,y_2$, the total number of which is $\\frac{1}{\\delta \\eta^2}$, completes the proof.\n\\end{proof}\n\nWe are now ready to prove Theorem \\ref{t:lln}. We start by showing the upper bound. Recall that $L_{\\alpha}(n)$ denotes the length of the constrained geodesic.\n\n\\begin{proposition}\n\\label{p:llnub}\nFix $\\varepsilon>0$. There exists $c=c(\\alpha,\\varepsilon)>0$ such that with probability at least $1-e^{-cn}$ we have \n$$L_{\\alpha}(n) \\leq 2(\\mathrm{w}_{\\alpha}+\\varepsilon)n.$$ \n\\end{proposition}\n\n\\begin{proof}\nRecall that for any increasing path $\\gamma,$ we denote its length or the number of points of the Poisson process it passes through by $|\\gamma|$.\nLet us now notice that $$|\\gamma|\\le \\sum_{i\\in I_{\\delta}}L\\biggl(\\bigl(i\\delta, \\gamma_{\\delta,\\eta}(i\\delta)\\bigr), \\bigl((i+1)\\delta, \\gamma_{\\delta,\\eta}((i+1)\\delta)+\\eta\\bigr)\\biggr).$$ \nThus by the previous lemma, with probability at least $1-e^{-cn},$ we have for all increasing paths, $\\gamma$ (from $(0,0)$ to $(1,1)$ in the rescaled space)\n\\begin{align*}\n|\\gamma|&\\le \\sum_{i\\in I_{\\delta}} 2\\delta \\sqrt{\\frac{\\gamma_{\\delta,\\eta}((i+1)\\delta)+\\eta)-\\gamma_{\\delta,\\eta}(i\\delta)}{\\delta}}n +2\\eta n, \\\\\n&= \\sum_{i\\in I_{\\delta}} 2n \\delta \\sqrt{\\dot \\gamma_{\\delta,\\eta}+\\frac{\\eta}{\\delta}} +2\\eta n.\n\\end{align*}\nAn argument involving truncating at $\\dot \\gamma_{\\delta,\\eta}< \\sqrt{\\frac{\\eta}{\\delta}}$ shows that, with probability at least $1-e^{-cn}$, for all increasing paths $\\gamma,$\n$$\\sum_{i\\in I_{\\delta}} n\\delta \\sqrt{\\dot \\gamma_{\\delta,\\eta}+\\frac{\\eta}{\\delta}}\\le n\\sum_{i} \\delta \\sqrt{\\dot \\gamma_{\\delta,\\eta}} +{\\left(\\frac{\\eta}{\\delta}\\right)}^{1\/4}O(n).\n$$\n\nThus, for $\\eta < \\delta^5$, with probability at least $1-e^{-cn},$ for all $\\gamma,$ \n\\begin{equation}\\label{approx34}\n|\\gamma|\\le n \\int_{0}^1 2\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x){\\rm{d}}x +O(\\delta n).\n\\end{equation} \n\nNow let $\\gamma$ be an \nincreasing path that traps area at least $(1\/2+\\alpha)$ (which corresponds to area $(1\/2+\\alpha)n^2$ in the unscaled model). By \\eqref{area14}, it follows that $\\int_0^1 \\gamma_{\\delta,\\eta}(x){\\rm{d}}x \\ge \\frac{1}{2}+\\alpha- O(\\delta)$ and hence, by continuity of $\\mathrm{w}_{\\alpha}$, we have $\\int_{0}^1 \\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x){\\rm{d}}x\\le ({\\rm w}_{\\alpha}+\\varepsilon\/2)$ by choosing $\\delta$ sufficiently small. By choosing $\\delta$ suitably small, this implies that, with probability at least $1-e^{-cn}$, the bound $|\\gamma|\\le 2( w_{\\alpha}+\\varepsilon)n$ for such increasing paths $\\gamma$. This completes the proof of the proposition.\n\\end{proof}\n\nWe now show the corresponding lower bound. \n\n\\begin{proposition}\n\\label{p:llnlb}\nFix $\\varepsilon>0$. There exists $c=c(\\alpha,\\varepsilon)>0$ such that, with probability at least $1-e^{-cn^2}$, we have \n$$L_{\\alpha}(n) \\geq 2(\\mathrm{w}_{\\alpha}-\\varepsilon)n.$$ \n\\end{proposition}\n\n \\begin{proof}\nThe derivation of this lower bound is straightforward: we will take a discretisation of $\\psi_{\\alpha}$ and then use the continuity of $\\dot \\psi$.\nFix $\\varepsilon>0$. Choose $\\theta>0$ and consider $\\psi_*=\\psi_{\\alpha+\\theta}$. Then, by definition, \n$\n\\int_{0}^1 \\psi_*(x){\\rm{d}}x\\ge \\frac{1}{2}+\\alpha+\\theta .\n$\nChoose $\\theta$ small enough that \n$\n{\\rm w}_{\\alpha+\\theta}=\\int_{0}^1 \\sqrt{\\dot \\psi_*(x)}{\\rm{d}}x\\ge {\\rm w}_{\\alpha}-\\frac{\\varepsilon}{2}.\n$\n\nBy using the continuity of $\\psi$ and $\\dot \\psi$, we see that, for given $\\varepsilon$, there exist sufficiently small choices of $\\delta,\\eta,$ such that the piecewise affine function $\\psi_{*,\\delta, \\eta}$ interpolating between the points $u_i:=(i\\delta,\\eta\\lfloor\\frac{\\psi_*(i\\delta)}{\\eta}\\rfloor)$\nsatisfies \n \\begin{align*}\n \\int_{0}^1 \\psi_{*,\\delta,\\eta}(x){\\rm{d}}x\\ge \\frac{1}{2}+\\alpha+\\frac{\\theta}{2},&\n \\int_{0}^1 \\sqrt{\\dot \\psi_{*,\\delta,\\eta}}(x){\\rm{d}}x\\ge {\\rm w}_\\alpha-\\frac{3\\varepsilon}{4}.\n \\end{align*}\nNow let $\\gamma_{(i,i+1)}$ \nbe the geodesic between the points $u_i$ and $u_{i+1}$. Let $\\gamma$ be the path obtained by concatenating the paths $\\gamma_{(i,i+1)}$ for $i=0,\\ldots,\\frac{1}{\\delta}-1$ \nUsing Lemma \\ref{uni100}, it follows that, with probability at least $1-e^{-cn^2}$, \n$$\n|\\gamma|\\ge \\sum_{i\\in I_{\\delta}} 2n\\delta \\sqrt{\\dot \\psi_{*,\\delta,\\eta}} -2n\\eta \\geq 2(\\mathrm{w}_{\\alpha}-\\varepsilon)n ,\n$$\nwhere we take $\\eta$ and $\\delta$ sufficiently small. Now by\nLemma \\ref{l:approx3}, we find that \n$$\\left |\\int_{0}^{1} \\gamma(x){\\rm{d}}x - \\int_{0}^1 \\psi_{*,\\delta,\\eta}(x){\\rm{d}}x \\right|\\le 2\\delta,$$ since $\\gamma$ and $\\psi_{*,\\delta,\\eta}$ agree on $I_{\\delta}$.\nHence, for $\\delta$ small enough,\n$\n\\int_{0}^1 \\gamma(x){\\rm{d}}x \\ge \\frac{1}{2}+\\alpha\n$.\nThus, $L_{\\alpha,n}$ is at least as large as $|\\gamma|$. Hence, we are done.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{t:lln}]\nCombining Proposition \\ref{p:llnub} and Proposition \\ref{p:llnlb}, we complete the proof of Theorem \\ref{t:lln} by noting that an upper tail bound of the form $\\mathbb{P}(L_{\\alpha}(n)\\geq k)\\leq e^{-k}$ for all $k\\geq 5n^2$ is easily obtained by bounding the upper tail of the Poisson-distributed total number of points. \n\\end{proof}\n\n\\subsection{Law of large numbers for the geodesic}\n\nWe now prove Theorem \\ref{lln2}. The proof is by contradiction: if ${\\rm dist}(\\Gamma_{\\alpha},\\psi_{\\alpha})\\ge \\Delta$, and the noise space is such that the events listed in Lemma \\ref{uni100}\noccur, then we would be able to construct an absolutely continuous function $h$ that traps area at least $1\/2+\\alpha$ and for which $\\int_0^1 \\sqrt{\\dot h}(x){\\rm{d}}x>\\int_0^1\\sqrt{\\dot \\psi_{\\alpha}}(x){\\rm{d}}x $. This would contradict extremality of $\\psi_{\\alpha}$.\nTo show the above inequality, we will employ a concavity argument. \nFor notational brevity, let $\\gamma=\\Gamma_{\\alpha}$ and $\\psi=\\psi_{\\alpha}$. For $\\delta,\\eta>0$, recall the definition of $\\gamma_{\\delta,\\eta}$ from the previous section. \nWe will consider the function $h=\\frac{\\gamma_{\\delta,\\eta}+\\psi}{2}$.\nFixing parameters $\\varepsilon,\\delta$ and $\\eta\\le \\delta^5,$\nwe will restrict attention to the event $\\mathcal{A}$,\non which:\n\\begin{enumerate}\n\\item $|\\gamma|\\le 2n \\int_{0}^1 \\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x){\\rm{d}}x +O(\\delta n),$ \\\\\n\\item $2({\\rm w}_\\alpha -\\varepsilon)n \\le |\\gamma|\\le 2({\\rm w}_\\alpha +\\varepsilon)n$.\n\\end{enumerate}\nBy Propositions \\ref{p:llnub} and \\ref{p:llnlb}, and \\eqref{approx34}, $\\mathcal{A}$ occurs with probability at least $1-e^{-cn}$. \nObserve that Proposition \\ref{solution1} implies that $\\dot \\psi$ is bounded away from zero and infinity on $[0,1]$. Thus, $\\dot \\psi$ lies in $[c_1,C_2]$ where $01$.\nObserving that $\\dot h=\\frac{\\dot \\gamma_{\\delta,\\eta}+{\\dot \\psi}}{2}$, \n we see that the concavity of $x\\to \\sqrt x$ implies that $\\sqrt{\\dot h}-\\frac{\\sqrt{\\dot \\gamma_{\\delta,\\eta}}+\\sqrt{\\dot \\psi}}{2},$ is non-negative. \nFurthermore, simple algebra shows that, for all $x\\in S$,\n\\begin{align}\\label{algebra101}\n\\sqrt{\\dot h}(x)-\\frac{[\\sqrt{\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]}{2} &= \\frac{(\\sqrt{\\dot \\psi}(x)-\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{4[\\sqrt{\\dot h}(x)+\\frac{[\\sqrt{\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]}{2}]}\\stackrel{\\eqref{bound303}}{\\ge} \\frac{(\\sqrt{\\dot \\psi}(x)-\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{8 C_2},\\\\\n\\nonumber\n&\\ge \\frac{({\\dot \\psi}(x)-{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{8C_2[\\sqrt {\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]^2}\\stackrel{\\eqref{bound303}}{\\ge} \\frac{({\\dot \\psi}(x)-{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{100 C_2^2}.\n\\end{align} \n\n\nThe proof is now completed by observing that a length gain has been realized:\\begin{align*}\n\\int_0^1 \\left[\\sqrt{\\dot h}(x)-\\frac{[\\sqrt{\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]}{2}\\right]{\\rm{d}}x& \\ge \\int_0^1 \\left[\\sqrt{\\dot h}(x)-\\frac{[\\sqrt{\\dot \\psi}(x)+\\sqrt{\\dot \\gamma_{\\delta,\\eta}}(x)]}{2}\\right] \\mathbf{1}(S){\\rm{d}}x \\\\\n&\\stackrel{\\eqref{algebra101}}{\\ge} \\int_0^1 \\frac{({\\dot \\psi}(x)-{\\dot \\gamma_{\\delta,\\eta}}(x))^2}{100 C_2^2}\\mathbf{1}(S){\\rm{d}}x \\\\\n&\\ge \\frac{1}{100C_2^2} \\left(\\int |({\\dot \\psi}(x)-{\\dot \\gamma_{\\delta,\\eta}}(x)|\\mathbf{1}(S){\\rm{d}}x \\right)^2\\\\\n &\\stackrel{\\eqref{algebra102}}{\\ge} \\frac{1}{2000C_2^2} \\Delta^2. \n\\end{align*}\nOn the event $\\mathcal{A},$ we see, by means of this inequality and the definition of $\\mathcal{A}$, that if $\\displaystyle{\\sup_x|\\gamma(x)-\\psi(x)|\\ge \\Delta} $ holds, \nthen $$\\int_0^1\\sqrt{\\dot h}(x){\\rm{d}}x \\ge [({\\rm w}_\\alpha-\\varepsilon)+\\frac{1}{C^2} \\Delta^2]$$ for some constant $C$ that depends only on $\\alpha$.\nAlso, both $\\int_{0}^1\\psi(x){\\rm{d}}x $ and $\\int_{0}^1\\gamma(x){\\rm{d}}x $ are at least $\\frac{1}{2}+\\alpha$ by definition, and hence by \\eqref{area14}, we have \n$$\n\\int_0^1 h(x){\\rm{d}}x =\\frac{\\int_0^1 \\psi(x){\\rm{d}}x +\\int_0^1 \\gamma_{\\delta,\\eta}(x){\\rm{d}}x }{2}\\ge \\frac{1}{2}+\\alpha-\\varepsilon\/2,\n$$ where $\\varepsilon$ can be made arbitrarily small by choosing $\\delta$ and hence $\\eta$ small enough. This inference contradicts the continuity of ${\\rm w}_{\\alpha}$ in $\\alpha$.\nHence we are done.\n\\qed\n\n\nWe now use similar arguments as those employed to prove the law of large numbers in order to establish a variant of Theorem \\ref{flat0} that is also uniform in $\\alpha$. This particular variant will be crucial in the proof of Theorem \\ref{t:mflub}.\n\n\n\\begin{theorem}\n\\label{flat1} \nFix $0< \\alpha_1<\\alpha_2 <\\frac{1}{2}$. For any small enough $\\delta>0$, there exist $\\gamma>0$ and $c>0$ such that, with probability at least $1-e^{-cn}$, all $\\delta$-interior facets of $\\Gamma_{\\alpha,n}$ for which $\\alpha\\in[\\alpha_1,\\alpha_2]$ make an angle with the $x$-axis that lies $(\\omega,\\pi\/2-\\omega)$.\n\\end{theorem}\n\n\\begin{proof} \nRecall that the proof of Theorem \\ref{flat0} \nused Theorem \\ref{lln2} and the strict convexity of the function $\\psi_\\alpha$. Since $\\psi_{\\alpha}$ is uniformly convex for all $\\alpha\\in [\\alpha_1,\\alpha_2]$, this proof will be complete, using the same arguments as in the proof of Theorem \\ref{flat0}, once we prove the following uniform version of Theorem \\ref{lln2}: for any $\\Delta>0$, with probability at least $1-e^{-cn}$, $$\\sup_{\\alpha\\in[\\alpha_1,\\alpha_2]}\\sup_x|\\Gamma_{\\alpha}(x)-\\psi_{\\alpha}(x)|\\le \\Delta,$$ where $c$ depends on $\\Delta$ and the interval $[\\alpha_1,\\alpha_2]$.\nWe fix an $\\varepsilon$ to be specified later and discretise the interval $[\\alpha_1,\\alpha_2]$ to obtain the set $\\mathcal{B}=\\{\\alpha_1, \\alpha_1+\\varepsilon, \\alpha_1+2\\varepsilon,\\ldots,\\alpha_2\\}$.\nFixing parameters $\\varepsilon,\\delta$ and $\\eta\\le \\delta^5$,\nagain as before we restrict attention to the event $\\mathcal{A}$ on which, for all $\\alpha \\in \\mathcal{B}$,\n\\begin{enumerate}\n\\item $|\\Gamma_{\\alpha}|\\le 2 n \\int_{0}^1 \\sqrt{\\dot{ \\Gamma}_{\\alpha,\\delta,\\eta}}(x){\\rm{d}}x +O(\\delta n)$; \n\\item and $2({\\rm w}_\\alpha -\\varepsilon)n \\le |\\Gamma_{\\alpha}|\\le 2({\\rm w}_\\alpha +\\varepsilon)n$.\n\\end{enumerate}\nSimilarly to a previous argument, by Propositions \\ref{p:llnub} and \\ref{p:llnlb} and \\eqref{approx34}, followed by a union bound, $P(\\mathcal{A})\\ge 1-e^{-cn}$. \nNow if $\\delta$ and $\\eta$ are chosen to be sufficiently small depending on $\\Delta,$ by the previous result, Theorem \\ref{lln2}, and a simple union bound, we obtain $$\\sup_{\\alpha\\in \\mathcal{B}}\\sup_x|\\Gamma_{\\alpha}(x)-\\psi_{\\alpha}(x)|\\le \\Delta.$$\nThe proof will proceed along the same lines as the proof of Theorem \\ref{lln2} did. With the aim of arriving at a contradiction, let $\\alpha\\in [\\alpha_1+i\\varepsilon,\\alpha_1+(i+1)\\varepsilon]$ be such that \n\\begin{equation}\\label{violate}\n\\sup_x|\\Gamma_{\\alpha}(x)-\\psi_{\\alpha}(x)|> \\Delta.\n\\end{equation}\nClearly, $|\\Gamma_{\\alpha_1+(i+1)\\varepsilon}|\\le |\\Gamma_{\\alpha}|\\le |\\Gamma_{\\alpha_1+i\\varepsilon}|$ holds by definition.\nSince $\\psi_{\\beta}$ is a continuous function of $\\beta$ in the supremum norm, we see that, for small enough $\\varepsilon$, $$\\displaystyle{\\sup_x|\\Gamma_{\\alpha}(x)-\\psi_{\\alpha_1+i\\varepsilon}(x)|\\ge \\Delta\/2}.$$\nNow, as we argued in the proof of Theorem \\ref{lln2}, this implies that \n\\begin{align}\\label{dev100}\n\\sup_x|\\Gamma_{\\alpha,\\delta,\\eta}(x)-\\psi_{\\alpha_1+i\\varepsilon}(x)|\\ge \\Delta\/2.\n\\end{align}\nNote then that\n$$\n|\\Gamma_{\\alpha}|\\ge |\\Gamma_{\\alpha_1+(i+1)\\varepsilon}|\\ge 2n({\\rm{w}}_{\\alpha_1+(i+1)\\varepsilon}-\\varepsilon) \\ge 2n({\\rm{w}}_{\\alpha}-\\varepsilon_1),\n$$ \nwhere where $\\varepsilon_1$ can be made small enough by choosing $\\varepsilon$ small enough. The first inequality follows by definition, and the second by the occurrence of $\\mathcal{A}$.\n\nThus, using \\eqref{approx34}, we find that \n$$\\int_{0}^1 \\sqrt{\\dot \\Gamma_{\\alpha,\\delta,\\eta}}(x){\\rm{d}}x\\ge {\\rm{w}}_{\\alpha}-\\varepsilon_1.$$ \nThis along with \\eqref{dev100} allows us to apply the concavity argument that appears in the proof of Theorem \\ref{lln2}: by choosing $\\varepsilon,\\delta,\\eta$ much smaller than $\\Delta$, we thus contradict the continuity of ${\\rm w }_{\\beta}$ at $\\beta=\\alpha$.\n\\end{proof}\n\n\n\\section{Lower Bound for Scaling Exponents}\n\\label{s:lb}\nIn this section we provide proofs of the lower bounds on ${\\rm MFL}(\\Gamma_n)$ and ${\\rm MLR}(\\Gamma_n)$, i.e., we prove Theorem \\ref{t:mfllb} and Theorem \\ref{t:mlrlb}. Most of the work goes into proving the ${\\rm MFL}$ lower bound Theorem~\\ref{t:mfllb}, since the lower bound for local roughness is a reasonably easy corollary of Theorem \\ref{t:tftail}. Let $\\alpha\\in (0,\\frac{1}{2})$ and $\\varepsilon>0$ be fixed for the rest of the section. Let $\\mathcal{A}_{\\varepsilon}$ denote the event that ${\\rm MFL}(\\Gamma_n)\\leq n^{3\/4-\\varepsilon}$. We shall show that the event $\\mathcal{A}_{\\varepsilon}$ is extremely unlikely. We start with an overview of the proof. We shall need a geometric definition. \n\n\n\\begin{definition}\n\\label{d:reg}\nLet $C>1,\\kappa>1$ be given constants. A sequence of points $u_0\\preceq u_1 \\preceq \\cdots \\preceq u_k$ is called a $(C,\\kappa)$-\\textbf{regular} sequence if the following conditions hold.\n\\begin{enumerate}\n\\item[(i)] The union of line segments joining $u_{i}$ to $u_{i+1}$ for $i=0,1,\\ldots,k-1,$ is convex.\n\\item[(ii)] The gradient of the all the line segments joining $u_i$ to $u_{i+1}$ is $\\in (\\frac{1}{\\kappa}, \\kappa)$.\n\\item[(iii)]The distance between the first and last point in the sequence i.e., $|u_k-u_0|\\in (\\frac{1}{C}n^{3\/4-\\varepsilon\/2}, Cn^{3\/4-\\varepsilon\/2})$.\n\\item[(iv)] The distance between the consecutive points in the sequence is small: $|u_i-u_{i+1}|\\leq n^{3\/4-\\varepsilon}$.\n\\item[(v)] Let $\\theta_1$ and $\\theta_2$ be the angles that the line segments $(u_0,u_1)$ and $(u_{k-1},u_{k})$ make with the positive $x$-axis (clearly $\\theta_1\\geq \\theta_2$ by hypothesis). Then $\\theta_1-\\theta_2 \\leq 100C^2 n^{-1\/4-\\varepsilon\/2}$.\n\\end{enumerate}\n\\end{definition}\n\nSee Figure \\ref{f:corner} for an illustration of this definition. \n\n\\begin{figure}[h] \n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[width=0.4\\textwidth]{regcorner.pdf} &\\includegraphics[width=0.3\\textwidth]{corner2.pdf} \\\\\n(a) & (b)\n\\end{tabular}\n\\caption{ (a) A sequence of regular corners $u_0,u_1,\\ldots, u_k$. The triangle $T$ is an isosceles triangle formed by vertices $u_0, u_k$ and $u_*$ such that the adjacent sides lie above the piecewise linear path passing through the points $u_0,u_1,\\ldots, u_k$. By definition of regularity, the triangle $T$ is contained in the parallelogram $R$ formed by the vertices $u_0, v_0,v_k$ and $u_k$ which has vertical height $n^{1\/2-\\varepsilon\/2}$. Observe that the height of this parallelogram is much smaller than the transversal fluctuation of paths between $u_0$ and $u_k$. \n(b) On $\\mathcal{A}_{\\varepsilon}$, there is a regular sequence of corners $u_0, u_1,\\ldots, u_{k}$ and $\\Gamma$ is the best constrained path. We show that there is an alternative path $\\gamma$ between $u_0$ and $u_1$ above the triangle $T$ that is extremely likely to be strictly larger in length than the restriction of $\\Gamma$ between $u_0$ and $u_{k}$. Then the path obtained by replacing $\\Gamma$ with $\\gamma$ between $u_0$ and $u_k$ has a larger length and traps a larger area, thus leading to a contradiction.\n}\n\\label{f:corner}\n\\end{figure}\n\n\nBefore proceeding let us try to motivate the item (v) of the above definition. Consider the least concave majorant of an increasing path from $(0,0)$ to $(n,n)$ as described in Definition \\ref{d:mlrmfl} \nThe total change of angle made by the line segments constituting the concave majorant with the $x$-axis is roughly around $\\pi\/2$ over a length of around $n$. So over a distance of around $n^{3\/4-\\varepsilon\/2}$, the change of angle should be around $n^{1\/4-\\varepsilon\/2}$ on average. The item (v) of the above definition asserts that on a regular sequence the change of angle is not much more than this average. \n\nThe following basic geometric consequence will be useful.\n\n\\begin{lemma}\n\\label{l:reggeo}\nLet $S=\\{u_0,u_1,\\ldots, u_k\\}$ be a $(C,\\kappa)$-regular sequence for some $C,\\kappa>1$. Let $L_{S}$ denote the union of line segments joining the consecutive points of $S$. Let $R$ denote the parallelogram with two vertical sides of length \n$n^{1\/2-\\varepsilon\/2}$ whose bottom side is the line segment joining $u_0$ and $u_k$; see Figure \\ref{f:corner}(a). Then there exists a point $u_*$ in $R$ such that the triangle $T$ formed by the points $u_0,u_k, u_*$ is an isosceles triangle (with the sides adjacent to $u_*$ being of equal length) contained in $R$ such that the two sides of $T$ adjacent to $u_*$ lie above the piecewise affine curve obtained by joining the consecutive points of $S$. \n\\end{lemma} \n\\begin{proof}\nConsider the angles $\\omega_1$ and $\\omega_2$ made by the line segments $(u_0,u_1)$ and $(u_{k-1},u_k)$ respectively with the line segment $(u_0,u_k)$. Also, let $v$ be the point of intersection obtained by extending the line segments $(u_0,u_1)$ and $(u_{k-1},u_k)$. By convexity, the piecewise affine curve obtained by joining the points of $S$ lies inside the triangle $(u_0,u_k,v)$. Moreover, it follows from elementary geometric arguments that $\\theta_1-\\theta_2=\\omega_1+\\omega_2$. Now let us consider the isosceles triangle $(u_0,u_k,u_*)$ where $(u_0,u_k)$ forms the base and $\\theta_1-\\theta_2$ is the value of the two equal angles. Since $\\theta_1-\\theta_2$ is at least as big as $\\omega_1$ and $\\omega_2$, it follows that the triangle $(u_0,u_k,v)$ is contained inside the triangle $(u_0,u_k,u_*)$. Moreover, the latter is clearly contained in a parallelogram of height $O(n^{3\/4-\\varepsilon\/2}n^{-1\/4-\\varepsilon\/2})$ where the constant in the $O(\\cdot)$ notation depends on $\\kappa$. Since for all large enough $n$, $n^{3\/4-\\varepsilon\/2}n^{-1\/4-\\varepsilon\/2}\\ll n^{1\/2-\\varepsilon\/2}$, we are done. \n\\end{proof}\n\nLet us now record the main steps of the proof of Theorem \\ref{t:mfllb}. Let $\\Gamma$ denote the constrained geodesic. Recall that $\\Gamma^{*}_{\\alpha,n}$ denotes least concave majorant of $\\Gamma$ and is an union of the facets as described in Definition \\ref{d:mlrmfl}. \n\n\\textbf{Step 1:} We shall fix a $(C,\\kappa)$-regular sequence $S=\\{u_0,u_1,\\ldots, u_k\\}$, and consider the best path~$\\gamma_{S}$ from $u_0$ to $u_{k}$ that passes through all the points in the sequence. We shall show that with very high probability the length of this path is much smaller than the length of the best path between $u_0$ and $u_{k}$. The reason that this event is likely is the following: because the lengths of the segments $\\{u_{i}, u_{i+1}\\}$ are much smaller than that of the segment $\\{u_{0}, u_{k}\\}$, it will turn out that the path $\\gamma$ will have a much smaller transversal fluctuation compared to the typical geodesic between $u_0$ and $u_{k}$. Thus, by Theorem~\\ref{t:constrained}, it is extremely likely that this path has a much smaller length. \n\n\\textbf{Step 2:} Next we will show that there exists a path between $u_0$ and $u_{k}$ that lies above the line segments joining the consecutive points of $S$ and whose length is comparable to that of the geodesic between $u_0$ and $u_{k}$. This will follow from the definition of regular sequence together with an application of Lemma \\ref{l:reggeo} and Theorem \\ref{baikrains}. \n\n\\textbf{Step 3:} Finally, we will show that, on the event $\\mathcal{A}_{\\varepsilon}$, it is overwhelmingly likely that there exists a $(C,\\kappa)$-regular sequence made out of consecutive corners of $\\Gamma^{*}_{\\alpha,n}$. Once we establish this, we will arrive at a contradiction: using the first two steps, we will construct a path that traps more area than does $\\Gamma$ and that also has a greater length.\n\n\nThe next proposition treats the first step.\n\n\\begin{proposition}\n\\label{p:step1}\nLet $S=\\{u_0,u_1,\\ldots, u_k\\}$ be a $(C,\\kappa)$-regular sequence. For $i=0,1,\\ldots,k-1,$ let $\\gamma_{i}$ be the geodesic\nbetween $u_{i}$ and $u_{i+1}$. Let $\\gamma_{S}$ denote the concatenation of the paths $\\gamma_{i}$. Then there exists $c>0$ such that, with probability at least $1-e^{-n^{c}}$, we have \n$$|\\gamma_{S}|\\leq \\mathbb{E} L(u_0,u_k)-n^{1\/4-\\varepsilon\/12}.$$\n\\end{proposition}\n\n\\begin{proof}\nLet $R'$ be the parallelogram with sides parallel to the sides of $R$ such that $u_0$ and $u_{k}$ are the midpoints of the vertical sides of $R'$; and the height of the vertical sides is $4n^{1\/2-\\varepsilon\/2}$. For each $i$, let $R_i$ denote the parallelogram with the following properties:\n\\begin{enumerate}\n\\item $R_i$ has one pair of vertical sides and the other pair of sides is parallel to the line segment joining $u_{i}$ and $u_{i+1}$. \n\\item The vertical sides have midpoints $u_{i}$ and $u_{i+1}$. \n\\item The height of the vertical sides is $n^{1\/2-3\\varepsilon\/5}$. \n\\end{enumerate} \nIt is straightforward to check from Definition \\ref{d:reg} that all the rectangles $R_{i}$ are contained in the rectangle $R'$. Now it follows from Theorem \\ref{t:tftail1} that, with probability at least $1-e^{-n^{c}}$, each geodesic $\\gamma_i$ is contained in the parallelogram $R_i$. In particular, on this event, \n$$ |\\gamma_{S}|\\leq L(u_0,u_{k}; R').$$\nThe result now follows using Theorem \\ref{t:constrained} and the assumed lower bound on $|u_0-u_{k}|$.\n\\end{proof}\n\n\n\n\\begin{figure}[h] \n\\centering\n\\begin{tabular}{ccc}\n\\includegraphics[width=0.4\\textwidth]{corner3.pdf} &\\includegraphics[width=0.4\\textwidth]{mlr.pdf} \\\\\n(a) & (b) \n\\end{tabular}\n\n\\caption{ (a) Illustrates the situation when all the facets are small. Using a priori bounds on the transversal fluctuations of the unconstrained geodesic, it follows that the segment of $\\Gamma_{\\alpha,n}$ between $u_0$ and $u_k$ lies in a thin parallelogram with width much smaller than the characteristic fluctuation scale. This then implies that the length of this segment is atypically low (see Theorem \\ref{t:constrained}).\n(b) Illustrates how in this situation one can replace the segment of $\\Gamma_{\\alpha,n}$ between $u_0$ and $u_k$ by a path that lies strictly above the parallelogram and fluctuates on a characteristic scale, and hence captures more area along with more length, which contradicts extremality of $\\Gamma_{\\alpha,n}$. This is used in the proof of Theorem \\ref{t:mlrub} } \n\\label{f:alternatepath20}\n\\end{figure}\nNext we move onto Step 2 where we show the existence of a ``good\" path between $u_0$ and $u_{k}$. We have the following proposition. \n\n\\begin{proposition}\n\\label{p:step2}\nLet $S=\\{u_0,u_1,\\ldots, u_k\\}$ be a $(C,\\kappa)$-regular sequence. Let $\\mathbb{L}_{S}$ denote the union of line segments joining $u_{i}$ to $u_{i+1}$.Then there exists $c>0$ such that, with probability at least $1-e^{-n^{c}}$, there exists a path $\\gamma_{*}$\nfrom $u_0$ to $u_{k}$ lying above $\\mathbb{L}_{S}$ such that\n$$|\\gamma_{*}| > \\mathbb{E} L(u_0,u_k)-n^{1\/4-\\varepsilon\/12}.$$\n\\end{proposition} \n\n\\begin{proof}\nWe will use the notation from Lemma \\ref{l:reggeo}. Let $u_*$ be the point in $R$ that satisfies the conclusion of that lemma.\nLet $\\gamma_{1}$ denote a path from $u_0$ to $u_{*}$ lying above the line segment joining $u_0$ and $u_{*}$ that achieves the length $L^{\\boxslash}(u_0,u_{*})$. Similarly, let $\\gamma_{2}$ denote a path from $u_{*}$ to $u_{1}$ lying above the line segment joining $u_*$ and $u_{1}$ that achieves the length $L^{\\boxslash}(u_*,u_{1})$. See Figure \\ref{f:corner} (b).\n\n\nLet $\\gamma_{*}$ denote the concatenation of the paths $\\gamma_1$ and $\\gamma_2$. Clearly $\\gamma_{*}$ lies above $\\mathbb{L}_{S}$. We shall show that, with overwhelming probability, $\\gamma_*$ also satisfies the other condition in the statement of the proposition. It follows by definition and some elementary geometry that both of the equal sides of the isosceles triangle $T$ have gradient in $(\\frac{1}{2\\kappa}, 2\\kappa)$, and also $|u_0-u_*|=|u_*-u_k|=\\Theta (n^{3\/4-\\varepsilon\/2})$. It follows using Theorem \\ref{baikrains} that with probability at least $1-e^{-n^{c}}$ one has \n$$ L^{\\boxslash}(u_0,u_*)\\geq \\mathbb{E} L(u_0,u_*)- n^{1\/4-\\varepsilon\/8}; \\qquad L^{\\boxslash}(u_*,u_k)\\geq \\mathbb{E} L(u_*,u_k)- n^{1\/4-\\varepsilon\/8}. $$\nAlso observe that $u_*$ is equidistant from $u_0$ and $u_k$ and that this point lies within the parallelogram $R$ whose height is $o(|u_0-u_k|^{2\/3})$. Thus, by a standard calculation, \n$$ \\mathbb{E} L(u_0,u_*)+ \\mathbb{E} L(u_*,u_k) -\\mathbb{E} L(u_0,u_k)= o(n^{1\/4-\\varepsilon\/6}).$$\nCombining these inferences, we find that, with probability at least $1-e^{-n^{c}}$, \n$$|\\gamma_{*}| > \\mathbb{E} L(u_0,u_k)-n^{1\/4-\\varepsilon\/8}.$$\nThis completes the proof of the proposition.\n\\end{proof}\n\nIn the final step, we establish that, on $\\mathcal{A}_{\\varepsilon}$, that is if ${\\rm MFL}(\\Gamma_{n})$ is smaller than $n^{3\/4-\\varepsilon}$, it is extremely likely that there exists a regular sequence whose points are consecutive corners of $\\Gamma^*_{\\alpha, n}$ (the least concave majorant of $\\Gamma_{\\alpha, n}$). \nLet ${\\rm Reg}(C,\\kappa)$ denote the event that there exist consecutive corners $u_0,u_1,\\ldots, u_{k}$ of $\\Gamma^*_{\\alpha, n}$ such that $\\{u_0,u_1,\\ldots,u_k\\}$ is a $(C,\\kappa)$-regular sequence. \n\n\\begin{proposition}\n\\label{l:hullreg} There exist $C,\\kappa$ such that, except for a sub-event of exponentially small probability, \nthe event $\\mathcal{A}_{\\varepsilon}$ is contained in the event ${\\rm Reg}(C,\\kappa)$ i.e. $$\\mathbb{P}(\\mathcal{A}_{\\varepsilon}\\setminus {\\rm Reg}(C,\\kappa)) \\le e^{-cn},$$ for some universal constant $c>0$.\n\\end{proposition}\n\n\n\nOn $\\mathcal{A}_{\\varepsilon}$, we shall find a regular sequence among interior facets. \nBefore proceeding we state an easy geometric lemma that will be used. The proof is provided in Section \\ref{es}.\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[scale=.4]{fig1.pdf}\n\\caption{Crude estimates on length of facets which subtend an angle $\\theta$ at the origin. }\n\\label{fig1}\n\\end{figure}\n\nFor any two points $p_1,p_2 \\in [0,n]^2$, denote by $\\theta(p_1,p_2)$ the acute angle between the lines joining the points $p_1$ and $p_2$ to the point $(n,0)$, i.e., the bottom right corner of the square (see Figure \\ref{fig1}).\n\\begin{lemma}\\label{crude1} There exists $\\theta_0$ such that with probability at least $1-e^{-cn}$ simultaneously for all $u,v$ on $\\Gamma^*_{\\alpha,n}$ with $\\theta(u,v)\\le \\theta_0$, then $|u-v|= \\Theta (\\theta(u,v) n)$. \n\\end{lemma} \n\n\\begin{proof}[Proof of Proposition \\ref{l:hullreg}]\nObserve that by definition, on $\\mathcal{A}_{\\varepsilon}$, any sequence $u_0\\preceq u_1 \\preceq \\cdots \\preceq u_k$ of consecutive corners of $\\Gamma^*_{\\alpha, n}$ satisfies condition (i) and (iv) of Definition \\ref{d:reg}. Fix $\\delta>0$. By Theorem \\ref{flat1}, with failure probability at most $e^{-cn}$ all the $\\delta-$interior facets have gradient $s$ in the interval $(1\/\\kappa, \\kappa)$ for some $\\kappa(\\delta)>1$. For the remainder of the proof, we shall assume that this event occurs. \nNow we partition all the $\\delta$-interior vertices of $\\Gamma^*_{\\alpha, n}$ (it is easy to see they are non-empty on $\\mathcal{A}_{\\varepsilon}$) into consecutive segments \n\\begin{equation}\\label{partition1}\n(v_{1},v_{2},\\ldots, v_{i_1}), (v_{i_1+1},v_{i_1+2}, \\ldots, v_{i_2}),\\ldots,\n\\end{equation} (where $v_i$'s are the consecutive corners on $\\Gamma^*_{\\alpha,n}$)\n such that \n\\begin{align*}\n\\theta(v_{i_j},v_{i_{j+1}-1}) &< n^{-1\/4-\\varepsilon\/2} \\le \\theta(v_{i_j+1},v_{i_{j+1}})\n\\end{align*}\nfor all $j=0,1,\\ldots,$ where $i_0=1$.\n\nWe denote the segment $(v_{i_j},v_{i_j+1}, \\ldots, v_{i_{j+1}})$ by $S_j$ \nThus if $S_1,S_2,\\ldots, S_{m}$ are all such segments, then $m=\\Theta(n^{1\/4+\\varepsilon\/2})$. It follows from convexity and the boundedness of the gradients (see Lemma \\ref{crude1} and Figure \\ref{fig1}) that $|v_{i_j}-v_{i_{j+1}}|=\\Theta (n^{3\/4-\\varepsilon\/2})$. Let $\\omega_{k}$ denote the angle made by the line segment $(v_{k}, v_{k+1})$ with the positive $x$-axis. For the sequence $S_j$, let $\\theta_{j}=\\omega_{i_j}-\\omega_{i_{j+1}-1}$. Notice that, by convexity, $\\theta_j$ is nonnegative for all $j$. Also observe that \n$$\\sum_{j} \\theta_{j}\\leq \\frac{\\pi}{2}.$$\nThis together with Markov's inequality and the fact that the number of $S_j$'s is $\\Theta(n^{1\/4+\\varepsilon\/2})$ implies that, for some constant $C$, there exists at least one $S_j$ such that $\\theta_j \\leq Cn^{-1\/4-\\varepsilon\/2}$. \nSetting $S=(u_0,u_1,\\ldots , u_{k})$ to be equal to such a sequence, we see that $S$ satisfies all conditions in the Definition \\ref{d:reg} for some $C,\\kappa>1$. This completes the proof of the proposition.\n\\end{proof}\n\n\n\nWith all this preparation, finally we are ready to prove Theorem \\ref{t:mfllb}.\n\n\\begin{proof}[Proof of Theorem \\ref{t:mfllb}] \nLet $\\alpha\\in (0,\\frac{1}{2})$ and $\\varepsilon>0$ be fixed as before. Let $(C,\\kappa)$ be such that the conclusion of Proposition \\ref{l:hullreg} holds. Let $\\mathcal{S}$ denote the event that there exists a $(C,\\kappa)$-regular sequence for which at least one of the conclusions in Proposition \\ref{p:step1} and Proposition \\ref{p:step2} does not hold.\nUsing the proofs of the said propositions, along with taking an union bound over all pairs of grid points obtained by coarse graining $[0,n]^2$ (exactly as Corollary \\ref{uniform1} followed from Theorem \\ref{t:moddev}), that $\\mathbb{P}(\\mathcal{S})\\leq e^{-n^{c}}$ for some $c>0$.\nBy Proposition \\ref{l:hullreg}, it suffices to show that \n$$\\mathcal{A}_{\\varepsilon}\\cap \\mathcal{S}^c \\cap {\\rm Reg}(C,\\kappa)=\\emptyset$$\nBy way of contradiction, let us assume that the set is non-empty. Fix a configuration in the set $\\mathcal{A}_{\\varepsilon}\\cap \\mathcal{S}^c \\cap {\\rm Reg}(C,\\kappa)$, and consider the least concave majorant $\\Gamma^*_{\\alpha, n}$ of the constrained geodesic. On $\\mathcal{A}_{\\varepsilon}\\cap {\\rm Reg}(C,\\kappa)$, there exists a $(C,\\kappa)$-regular sequence $S=\\{u_0,u_1,\\ldots , u_{k}\\}$ consisting of consecutive corners of $\\Gamma^{*}_{\\alpha,n}$. On $\\mathcal{S}^c$, there exists a path $\\gamma_*$ from $u_0$ to $u_{k}$ satisfying the conclusion of Proposition \\ref{p:step2}. Let $\\gamma_{0}$ denote the restriction of $\\Gamma$ between $u_0$ and $u_{k}$. Clearly, \n$$ |\\gamma_{0}|\\leq \\sum_{i} L(u_i,u_{i+1}).$$ It follows from Propositions \\ref{p:step1} and \\ref{p:step2} that, on $\\mathcal{S}^c$, $|\\gamma_{0}|< |\\gamma_{*}|$. Also observe that $\\gamma_{0}$ lies below the line segments joining $u_{i}$ to $u_{i+1}$ whereas $\\gamma_{*}$ lies above these segments. Hence the increasing path $\\Gamma'$ from $(0,0)$ to $(n,n)$ that coincides with $\\Gamma$ between $(0,0)$ and $u_0$ and between $u_{k}$ and $(n,n)$ and coincides with $\\gamma_{*}$ between $u_0$ and $u_{k}$ traps at least as much area as does $\\Gamma$. However, by the above discussion, $|\\Gamma'|> |\\Gamma|$. This contradicts the extremality of $\\Gamma$ and completes the proof of the theorem. \n\\end{proof}\n\n\\subsection{Proof of lower bound of ${\\rm MLR}$}\nIn this subsection we prove Theorem \\ref{t:mlrlb}. That is, we show that the maximum local roughness of the facets has scaling exponent at least $\\frac{1}{2}$. The basic idea of the proof is to use KPZ scaling to argue that, if there is a facet of length roughly $n^{3\/4}$, then along that facet, the local roughness is likely to be at least $(n^{3\/4})^{2\/3}=n^{1\/2}$. We now make this more precise.\n\nFix $\\varepsilon>0$. We have just proved that, with high probability, there exists a facet of length at least $n^{3\/4-\\varepsilon}$. In fact, by the proof of Theorem \\ref{t:mfllb}, more is true. Fix $\\kappa\\gg 1$ a large constant. Let $\\mathcal{B}_{\\varepsilon}=\\mathcal{B}_{n,\\varepsilon,\\kappa}$ denote the event that $\\Gamma^{*}_{\\alpha,n}$ (see Definition \\ref{d:mlrmfl}) has a facet of length at least $n^{3\/4-\\varepsilon}$ with endpoints $v_0$ and $v_{1}$ such that the straight line joining $v_0$ and $v_1$ has gradient $\\in (1\/\\kappa, \\kappa)$. The following lemma is a consequence of the proof of Theorem \\ref{t:mfllb}. \n\n\\begin{lemma}\n\\label{l:be}\nFor all large enough $\\kappa$ (depending on $\\alpha$),\n$\\mathbb{P}(\\mathcal{B}_{\\varepsilon})\\ge 1-e^{-n^c}$ for all $n$ sufficiently large, \nwhere $c$ is a constant depending only on $\\varepsilon$ and $\\alpha$.\n\\end{lemma}\n\n\\begin{proof}\nNote that the proof of Theorem \\ref{t:mfllb} in fact showed that there must be a $\\delta$-interior facet of length at least $n^{3\/4-\\varepsilon}$ with sufficiently large probability. Since all the $\\delta$-interior facets satisfy the gradient requirement (see Theorem \\ref{flat1}), we are done. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{t:mlrlb}]\nLet $\\mathcal{C}_{\\varepsilon}$ denote the event that ${\\rm MLR}(\\Gamma_n) \\leq n^{1\/2-\\varepsilon}$. Thus, to prove Theorem~\\ref{t:mlrlb}, it suffices to show that $$\\mathbb{P}(\\mathcal{C}_{\\varepsilon}\\cap \\mathcal{B}_{\\varepsilon})\\le e^{-n^c}.$$ \n\nLet $\\Gamma$ denote the constrained geodesic. On $\\mathcal{C}_{\\varepsilon}\\cap \\mathcal{B}_{\\varepsilon}$, let $v_0$ and $v_1$ be corners of $\\Gamma^*_{\\alpha, n}$ satisfying the conditions in the definition of $\\mathcal{B}_{\\varepsilon}$. Consider the restriction $\\Gamma(v_0,v_1)$ of $\\Gamma$ between $v_0$ and $v_1$. As ${\\rm MLR}(\\Gamma) \\leq n^{1\/2-\\varepsilon}$; it follows that there exists $C=C(\\kappa)>0$ such that the restriction of $\\Gamma$ (call it $\\gamma$) between $v_0$ and $v_1$ is contained in the parallelogram $U$ that has a pair of vertical sides of height $Cn^{1\/2-\\varepsilon}$ and whose top side is the line segment joining $v_0$ and $v_1$. Now observe that the height of the parallelogram is much smaller than the on scale transversal fluctuation of geodesics between $v_0$ and $v_1$, these fluctuations being at least $$(n^{3\/4-\\varepsilon})^{2\/3-\\varepsilon\/100}\\ge n^{1\/2-{3\\varepsilon}\/{4}} . $$ \nThus, Corollary \\ref{uniform0} implies that the bound \n$$\\mathbb{E}(L(v_0, v_1))-|\\gamma| \\ge |v_0-v_1|^{1\/3- 5\\varepsilon\/9}$$\nfails with probability at most $e^{-n^{c}}$. By Corollary \\ref{uniform2}, \n$$\n L^{\\boxslash}(v_0,v_1)-\\mathbb{E}(L(v_0,v_1)) \\ge -|v_0-v_1|^{\\frac{1}{3}+\\varepsilon\/100}\n $$\nexcept, again, on a set of probability at most $e^{-n^{c}}$.\nConsider now the path $\\Gamma'$ that agrees with $\\Gamma$ outside the segment $(v_0,v_1)$ and that between $v_0$ and $v_1$ equals $\\gamma^{\\boxslash}(v_0,v_1)$, which recall is a path from $v_0$ to $v_1$ lying above the line segment that joins these two points and achieves length $L^{\\boxslash}(v_0,v_1)$.\nWe see then that $\\Gamma'$ has greater length, and traps as much area as,~$\\Gamma$. \nThis contradicts the assumption of length maximality for $\\Gamma$; (similarly to the proof of Theorem \\ref{t:mfllb}. See Figure \\ref{f:alternatepath20}(b).)\n\\end{proof}\n\n\\section{Upper Bound for Scaling Exponents}\n\\label{s:ub}\nThis section is devoted to the proof of Theorem \\ref{t:mflub}. That is, it will be shown here that, for a dense set of $\\alpha$ that may depend on $n$, the maximum facet length of the constrained geodesic is very likely to be at most \n$n^{3\/4+o(1)}$. Theorem \\ref{t:mlrub} will then follow by the KPZ scaling enjoyed by transversal fluctuations. \n\n\nWe explain the main ideas before giving proofs. Recall Definition \\ref{good10}. The basic idea is to show, using Corollary \\ref{uniform2}, \nthat if a certain value of $\\alpha$ is not `good' (recall Definition \\ref{good10}),\nwe may perform a ``landgrab\" operation. More formally let us assume that there is a facet $(v_0,v_1)$ which has length $n^{3\/4+\\varepsilon}$ and has a reasonable gradient bounded away from zero and infinity. Theorem \\ref{baikrains} allows us to find a path $\\gamma$ from $v_0$ to $v_1$ which lies above the facet $(v_0,v_1)$ at a characteristic height of $(n^{3\/4+\\varepsilon})^{2\/3}$ and has characteristic fluctuations (we have room to allow some error) i.e. by Corollary \\ref{uniform2} we can assume \n$$|{\\gamma}|\\ge \\mathbb{E}(L(v_0,v_1)) -(n^{3\/4+\\varepsilon})^{1\/3+\\varepsilon\/1000}\\ge \\mathbb{E}(L(v_0,v_1))-n^{1\/4+2\\varepsilon\/5}.$$\nSimilarly by an Corollary \\ref{uniform1} we can assume that $$|\\Gamma(v_0,v_1)| \\le \\mathbb{E}(L(v_0,v_1))+ n^{1\/4+2\\varepsilon\/5}$$ \nwhere $\\Gamma(v_0,v_1)$ is the restriction of the constrained geodesic $\\Gamma$ between $v_0$ and $v_1$. \nThus the path $\\Gamma'$ which is obtained from $\\Gamma$ by replacing $\\Gamma(v_0,v_1)$ by $\\gamma$ traps at least $n^{5\/4+{5\\varepsilon}\/{3}}$ more area than does $\\Gamma$ and loses at most $n^{1\/4+2\\varepsilon\/5}$ in length. Repeating this operation about $n^{3\/4-5\\varepsilon\/3}$ times (we can do that provided there are no good $\\alpha$ in this interval), one gains an area that is $\\Theta(1)$ while losing at most $O(n^{1-\\varepsilon})$ in length with very high probability. This contradicts the fact that $\\mathbb{E} L_{\\alpha_1}(n) -\\mathbb{E} L_{\\alpha_2}(n)=\\Theta(n)$ for $\\alpha_1<\\alpha_2$ (an easy consequence of Theorem \\ref{t:lln} using ${\\rm w}_{\\alpha}$ is strictly decreasing) using Theorem \\ref{concentration1}. Next we provide the details needed to make the above argument precise. \n\nFix $0<\\alpha_1 <\\alpha_2 <\\frac{1}{2}$ and also $\\varepsilon>0$ and $\\delta\\in (0,\\pi\/4)$. Let $\\mathcal{G}_{\\alpha_1,\\alpha_2}$ denote the event that no $\\alpha\\in [\\alpha_1,\\alpha_2]$ is $(n,\\varepsilon, \\delta)$-good. Let $B_{n}=B_{n,\\varepsilon,\\kappa}$ be the set of all pairs $(x,y)\\in [0,n]^2$ satisfying the following conditions. \n\\begin{enumerate}\n\\item $x\\preceq y$.\n\\item The straight line joining $x$ and $y$ has gradient $\\in (\\frac{1}{\\kappa}, \\kappa)$.\n\\item $|x-y|\\geq n^{3\/4}$. \n\\end{enumerate} \nFor $(x,y)\\in B_n$, let $A_{x,y}$ denote the event that $|L^{\\boxslash}(x,y)-\\mathbb{E} L(x,y)|\\leq |x-y|^{1\/3+\\varepsilon\/1000}$ and $|L^{*}(x,y)-\\mathbb{E} L(x,y)|\\leq |x-y|^{1\/3+\\varepsilon\/1000}$ where $L^*(x,y)$ denotes the best path between $x$ and $y$ that is constrained to stay below the straight line joining $x$ and $y$. Let $\\mathcal{A}_{\\varepsilon}$ denote the event that $A_{x,y}$ holds for all $(x,y)\\in B_n$. Further let $S_{\\delta}$ denote the event in the statement of Theorem \\ref{flat1}, i.e., all the $\\delta$ interior facets of $\\Gamma^*_{\\alpha, n}$ has moderate gradients for all $\\alpha\\in [\\alpha_1, \\alpha_2]$. We have the following proposition.\n\n\\begin{figure}[h]\n\\center\n\\includegraphics[width=0.3\\textwidth]{landgrab.pdf}\n\\caption{The landgrab operation. By replacing the path $\\Gamma$ by the path $\\gamma$ one gains at least the amount of the area in the shaded region, whereas the event $A_{\\varepsilon'}$ ensures that the length loss is not too much.}\n\\label{f:landgrab}\n\\end{figure}\n\n\n\\begin{proposition}\n\\label{p:landgrab}\nThere exists $\\kappa=\\kappa(\\delta,\\varepsilon)$ sufficiently small such that, on $\\mathcal{A}_{\\varepsilon}\\cap S_{\\delta}\\cap \\mathcal{G}_{\\alpha_1,\\alpha_2}$, there exists $\\alpha_* \\geq \\alpha_2$ for which \n$$L_{\\alpha_1}(n)-L_{\\alpha_*}(n)\\leq (\\alpha_{*}-\\alpha_1) n^{1-6\\varepsilon\/5}.$$\n\\end{proposition}\n\n\\begin{proof}\nWe perform the following recursive construction. Let $\\beta_0=\\alpha_1$. For $\\beta_{i}\\in [\\alpha_1,\\alpha_2]$ construct $\\beta_{i+1}$ recursively as follows: find the longest $\\delta$-interior facet in $\\Gamma_{\\beta_i,n}$. On $\\mathcal{G}_{\\alpha_1,\\alpha_2}$, the longest $\\delta$-interior facet $(x_i,y_i)$ has length at least $n^{3\/4+\\varepsilon}$. Note that on $S_{\\delta}$, the endpoints $x_{i}$ and $y_{i}$ of this facet satisfy $(x_{i},y_{i})\\in B_n$ for some $\\kappa=\\kappa(\\delta)$. Now pick a point $z_{i}$ at orthogonal distance $|x_{i}-y_{i}|^{2\/3}$ from the midpoint of the line segment joining $x_i$ and $y_{i}$; (see Figure \\ref{f:landgrab}). Thus the area of the triangle $(x_i,y_i,z_i)$ is at least $c|x_{i}-y_{i}|^{5\/3}$ for some constant $c$ that does not depend on $i$. Set $\\beta_{i+1}=\\beta_{i}+ cn^{-2}|x_{i}-y_{i}|^{5\/3}$. Now consider the path $\\gamma$ that coincides with $\\Gamma_{\\beta_{i},n}$ outside the facet $(x_i,y_i)$, and is formed by concatenating $\\gamma^{\\boxslash}(x_i,z_i)$ and $\\gamma^{\\boxslash}(z_i,y_i)$ between $x_i$ and $y_{i}$. Clearly the area under the curve $\\gamma$ is at least $(\\frac{1}{2}+\\beta_{i+1})n^2$. Also, on $A_{\\varepsilon}$, \n$$|\\Gamma_{\\beta_{i},n}|-|\\gamma|\\leq |x_{i}-y_{i}|^{1\/3+\\varepsilon\/500}.$$\nIt follows that \n$$L_{\\beta_i}(n)-L_{\\beta_{i+1}}(n) \\leq |x_{i}-y_{i}|^{1\/3+\\varepsilon\/500}.$$\n\n\n\nDenote $\\alpha_{*}=\\beta_{i_0}$, where $i_0$ is the smallest index $i$ for which $\\beta_{i}\\geq \\alpha_2$. It follows that \n$$L_{\\alpha_1}(n)-L_{\\alpha_*}(n)\\leq \\sum_{i=0}^{i_0-1} |x_{i}-y_{i}|^{1\/3+\\varepsilon\/500}\\leq c^{-1}(\\alpha_*-\\alpha_1)n^{2} \\left(\\max_{i\\leq i_0} |x_i-y_i|^{-4\/3+\\varepsilon\/500}\\right).$$\nSince $|x_i-y_{i}|\\geq n^{3\/4+\\varepsilon}$ for all $i$, it follows that the final term is at most $n^{-1-4\\varepsilon\/3}$ and that completes the proof of the proposition.\n\\end{proof}\n\nWe may now complete the proof of Theorem \\ref{t:mflub}. \n\n\\begin{proof}[Proof of Theorem \\ref{t:mflub}]\nLet $\\kappa$ be large enough that the conclusion of Proposition \\ref{p:landgrab} holds. Using Theorem \\ref{flat1} and Corollary \\ref{uniform}, it follows that, for this choice of $\\kappa$ (and for given $\\alpha_1,\\alpha_2, \\delta$ and $\\varepsilon$), one has that, for some constant $c>0$,\n$$\\mathbb{P}[\\mathcal{A}_{\\varepsilon}^{c}\\cup S_{\\delta}^{c}] \\leq e^{-n^c}.$$\nHence, using Proposition \\ref{p:landgrab}, it suffices to show that \n\\begin{equation}\n\\label{e:suffice}\n\\mathbb{P}\\left (L_{\\alpha_1}(n)-L_{\\alpha_2}(n)\\leq (1\/2-\\alpha_1)n^{1-6\\varepsilon\/5}\\right )\\leq e^{-n^c}\n\\end{equation}\nfor some $c>0$. Notice that in the above step we have used the trivial inequality $L_{\\alpha_2}(n)\\geq L_{\\alpha_*}(n)$ for all $\\alpha_{*}\\in [\\alpha_2,\\frac{1}{2})$.\nIt follows now from Theorem \\ref{t:lln} that $\\mathbb{E} L_{\\alpha_1}(n)-\\mathbb{E} L_{\\alpha_2}(n)\\geq \\frac{{\\rm w}_{\\alpha_1}-{\\rm w}_{\\alpha_2}}{2}n$ for $n$ sufficiently large. The claimed bound \\eqref{e:suffice} now follows from Theorem \\ref{concentration1} and the fact that ${\\rm w}_{\\alpha_1}-{\\rm w}_{\\alpha_2} >0$. This completes the proof.\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{t:mlrub}}\nThis derivation is similar to that by which Theorem \\ref{t:mlrlb} follows from Theorem \\ref{t:mfllb}. \nBy Theorem \\ref{flat1}, for all $\\alpha \\in [\\alpha_1,\\alpha_2]$ and all small enough $\\delta$, there is a probability at least $1-e^{-cn}$ that all the $\\delta$-interior facets of $\\Gamma^*_{\\alpha,n}$ are $\\kappa-$ steep for some value of $1\\le \\kappa<\\infty$.\nNow for any good $\\alpha\\in [\\alpha_1, \\alpha_2]$ (note that the set of good $\\alpha'$s is a function of the underlying point process and hence is random), \nby definition, all the interior facets of $\\Gamma^*_{\\alpha,n}$ have length at most~$n^{3\/4+\\varepsilon}$. Moreover, by the uniform bounds discussed above we may restrict to the case where all the facets are $\\kappa$-steep. Suppose that the maximum local roughness restricted to the $\\delta$-interior facets is at least $n^{1\/2+\\varepsilon}$. Let $u$ and $v$ be the endpoints of the facet that attains this maximum local roughness. Let $\\gamma(u,v)$ be the segment of $\\Gamma_{\\alpha,n}$ between $u$ and $v$. \nBy using Corollary \\ref{uniform0}, we see that, with probability at least $1-e^{-n^c}$, \n$$|\\gamma(u,v)|< \\mathbb{E}(L(u,v))- S(u,v).$$ On the other hand,\n by Corollary \\ref{uniform2} again, we find that, except on an event of probability at most $e^{-n^c}$,\n$$L^{\\boxslash}(u,v)\\ge \\mathbb{E}(L(u,v)))- S(u,v).$$\nThus, the path that agrees with $\\Gamma$ from $(0,0)$ to $u$ and from $v$ to $(n,n)$ \nand equals the path $\\gamma^{\\boxslash}(u,v)$ between the points $u$ and $v$ clearly captures at least as much area as does $\\Gamma_{\\alpha,n}$ and also contains more points than does $\\Gamma_{\\alpha,n}$. This contradicts the extremality of the latter path. \n \\qed\n\n\\subsection{Possible extensions and difficulties}\\label{areafluc} \n We end this section with a few remarks related to the first two open problems mentioned in Section \\ref{oq}. A natural approach to quantifying the proof of Theorem \\ref{t:mflub} is via Proposition \\ref{p:landgrab}. Namely, one can hope to bound below the density of those $\\alpha\\in [\\alpha_1,\\alpha_2]$ that are $(n,\\varepsilon)$ good (where here we ignore the parameter $\\delta$ for the purpose of illustration.) This is because using similar arguments to those used in deriving these two results, one can hope to prove the following uniform bound: with high probability the noise space is such that for all $\\alpha\\in [\\alpha_1,\\alpha_2]$ we have $L_{\\alpha}(n)-L_{\\alpha_*}(n)\\le \\tilde O(\\ell^{1\/3})$ for $\\alpha_*=\\alpha+ \\ell^{5\/3}n^{-2},$ where $\\ell=\\max({\\rm{MFL}}(\\Gamma_{\\alpha,n}),n^{3\/4}),$ and where the $\\tilde O$ notation hides poly-logarithmic multiplicative factors (which arise due to the exceptional nature of the facet endpoints that contributes a polynomial entropy factor). Thus, the above implies that the length-to-area gradient in scaled coordinates (with length measured in units of $n$ and area measured in units of $n^2$) is $\\tilde O(\\frac{n}{\\ell^{4\/3}})=\\tilde O(1)$ as by definition $\\ell \\ge n^{3\/4}$. Also, we know that, for all $\\alpha$ which are\n $(n,\\varepsilon)$ bad, the gradient is polynomially small (since by the land-grab argument (see Figure \\ref{f:landgrab}), the length loss is much smaller than the area gain). This coupled with the fact that $L_{\\alpha_1}-L_{\\alpha_2}=\\Theta(n),$ should then allow to conclude that the density of good $\\alpha$ is at least $\\frac{1}{\\tilde O(1)}$.\n\nNotice that our approach for proving upper bounds relies on the joint coupling of the process for various values of $\\alpha$. This is quite different in spirit from the known proofs of similar statements in other contexts such as phase separation \\cite{AH1} where the upper bound was proven for a fixed value of $\\alpha$. The key ingredient there was an understanding of the excess area fluctuation (see Open question (2) in Section \\ref{oq}) which then allowed a resampling argument to work. However, since we do not have such bounds, we have to work across various values of $\\alpha$ simultaneously. \n\n A heuristic argument in our setting proving an upper bound of $n^{5\/4+o(1)}$ on the excess area trapped by $\\Gamma=\\Gamma_{\\alpha}$, assuming that $\\alpha$ is $(n,\\varepsilon\/10)$ good (recall Definition \\ref{good10}) can be made as follows: Suppose that the excess area is more than $n^{5\/4+\\varepsilon}$. Since, by hypothesis the longest facet length is at most $n^{3\/4+\\varepsilon\/10}$ , we may find two points $x$ and $y$ on $\\Gamma^*_{\\alpha,n}$ such that $|x-y|\\approx n^{3\/4+\\varepsilon\/5}$ and segment of $\\Gamma^*_{\\alpha,n}$ between $x$ and $y$ has average curvature (the average distance of the curve from the line segment joining $x$ and $y$ is what it should be on for a circle; see Figure \\ref{def} (b)). Now consider shortcutting $\\Gamma$ between $x$ and $y$ and hence replacing the subpath of $\\Gamma_{\\alpha,n}$ between $x$ and $y$ by the unconstrained geodesic between $x$ and $y$. This operation will create an area loss (approximately $n^{5\/4+3\\varepsilon\/5}$) which is not enough to violate the area constraint, because the excess area before the shortcut was at least $n^{5\/4+\\varepsilon}$. Moreover, the shortcut increases the overall weight and hence contradicts extremality of $\\Gamma_{\\alpha,n}$.\n\n\\section{Proofs of some of the earlier statements}\\label{es}\nIt remains to provide proofs of Lemmas \\ref{l:exist},\\ref{l:ac}, \\ref{decreasing1}, \\ref{crude1} and Theorem \\ref{concentration1} that were postponed.\n\n\n\\subsection{Proof of Lemma \\ref{l:exist}} This subsection follows closely the arguments presented in \\cite[Section 3]{DZ2}. Let $\\phi_1,\\phi_2,\\ldots$ be a sequence of elements in $\\mathcal{B}_{\\alpha}$ such that $$\\lim_{n}J(\\phi_n)=J_{\\alpha}.$$ \nWe equip the space of bounded signed measures on $[0,1]$ with the weak topology generated by $\\mathcal{C}=C[0,1]$ (the space of continuous functions on $[0,1]$). \nSince for all $n\\ge1,$ $\\phi_n$ correspond to sub-probability measures, using compactness, by passing to a subsequence (denoted again by $\\{n\\}$ for convenience) let $\\phi_n$ converge to $\\psi\\in \\mathcal{M},$ in the weak topology. \nThis in particular implies $\\phi_n(x)$ converges to $\\psi(x)$ almost everywhere (since the convergence happens at all continuity points of $\\psi$, see e.g., \\cite[Section 3.2]{Dur} ). Since $\\phi_n$'s are all bounded by $1$ and belong to $\\mathcal{B}_{\\alpha},$ by the bounded convergence theorem, it follows that $\\psi \\in \\mathcal{B}_{\\alpha}$. \nNext we will show that \n\\begin{equation}\\label{claim203}\n\\lim_{n}J(\\phi_n)\\le J(\\psi).\n\\end{equation}\nClearly this implies $J(\\psi)=J_{\\alpha}$ and hence we would be done.\nThe claim in \\eqref{claim203} follows from the upper semicontinuity of $J(\\cdot)$ or equivalently the lower semicontinuity of $-J(\\cdot)$.\nTo show the latter one represents $-J(\\cdot)$ as an appropriate Legendre transform. \nTo proceed we need some definitions: Clearly \\eqref{variational1} extends naturally to all non-negative measures. We extend $J$ to all of $\\mathcal{M}$ by setting it to be $\\infty$ for all measures which are not non-negative. \nMoreover, for any $f\\in \\mathcal{C},$ define\n $$\\Lambda(f)=\\left \\{ \\begin{array}{cc}\n\\infty & \\text {if } \\int_0^1\\frac{1}{|f(s)|}ds =\\infty \\text{ or if } f\\ge 0 \\text{ on a set of positive Lebesgue measure}. \\\\\n-\\frac{1}{4} \\int_{0}^1 \\frac{1}{f(s)}ds & \\text{ otherwise}\n\\end{array}\n\\right.\n$$\nNow for any $\\phi \\in \\mathcal{B},$ define $\\Lambda^*(\\phi)=\\displaystyle{\\sup_{f\\in C[0,1]} \\left[\\int_{0}^1 f d(\\phi) -\\Lambda(f)\\right]}$. \nThis function, by definition, is lower semicontinuous, since for any $f\\in C[0,1],$ and a sequence $\\phi_n$ converging to $\\phi,$ by definition of weak convergence, $$\\lim_{n\\to \\infty}\\int_0^1 f d(\\phi_n) = \\int_0^1 f d(\\phi).$$ Thus the proof is complete by \\cite[Lemma 5]{DZ2} which says $\\Lambda^*(\\phi)=-J(\\phi)$. \n\\qed\n\n\n\\subsection{Proof of Lemma \\ref{l:ac}}\nAs already hinted right after the statement of Lemma \\ref{l:ac}, the proof is by contradiction. Assuming that the singular part of $\\psi,$ which we denote by $\\psi_s$ (see \\eqref{decomposition23}). has positive mass, we create a modification $\\psi_1$ such that $J(\\psi_1)> J(\\psi)$ (see \\eqref{variational1}). However in the process, it is possible that we violate the area constraint, i.e. $\\int_0^1\\psi_1(x){\\rm{d}}x < \\frac{1}{2}+\\alpha$. Thus we make a second modification to construct a function $\\psi_2$ with the property $J(\\psi_2)> J(\\psi),$ and moreover \nit satisfies the area constraint as well. This contradicts the extremality of $\\psi$ which was a part of the hypothesis. The details follow.\n\nLet us assume that $\\psi_s$ has total mass $k>0$. Then clearly, without loss of generality, we can assume $\\psi_s=k\\delta_0$, i.e., there is an atom of mass $k$ at zero, since \n\\begin{align*}\n\\int_{0}^1 (\\psi_{ac}(x)+k) {\\rm{d}}x \\ge \\int_{0}^1 (\\psi_{ac}(x)+\\psi_{s}(x)) {\\rm{d}}x\\ge \\frac{1}{2}+\\alpha. \n \\end{align*}\nAlso, because the absolutely continuous part stays the same, $J(\\psi_{ac}+k\\delta_0)=J(\\psi)$. \n We will now make some local modifications to contradict extremality of $\\psi$. \n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[scale=.7]{fig4.pdf}\n\\caption{Local improvement}\n\\label{fig9}\n\\end{figure}\n\nLet $r>0$ be a small number to be chosen later. Consider a new function $\\psi_1$, as illustrated in Figure \\ref{fig9}, which is a linear function interpolating $(0,0)$ and $(r,\\psi(r))$ and which agrees with $\\psi$ on $[r,1]$. Since $\\psi(r)=\\psi_{ac}(r)+k$,\nit follows that\n\\begin{equation}\\label{arealoss}\n\\int_{0}^1 \\psi_1(x){\\rm{d}}x \\ge \\int_{0}^1 \\psi(x){\\rm{d}}x -\\frac{1}{2}kr-O(\\psi_{ac}(r)r).\n\\end{equation}\nOn the other hand, for all small enough $r$,\n\\begin{align}\\label{lengthgain}\n\\int_{0}^1 \\sqrt{\\dot \\psi_1(x)} {\\rm{d}}x &\\ge \\int_{0}^1 \\sqrt{\\dot\\psi(x)} {\\rm{d}}x -\\int_{0}^r \\sqrt{\\dot\\psi(x)} {\\rm{d}}x + \\sqrt{kr}\\\\\n\\nonumber\n&\\ge \\int_{0}^1 \\sqrt{\\dot\\psi(x)} {\\rm{d}}x -\\sqrt{\\psi_{ac}(r) r}+ \\sqrt{kr},\\\\\n\\nonumber\n& \\ge \\int_{0}^1 \\sqrt{\\dot\\psi(x)} {\\rm{d}}x + \\frac{\\sqrt{kr}}{2}.\n\\end{align}\nThe second last inequality uses the bound $\\int_0^r \\sqrt{\\dot \\psi(x)} {\\rm{d}}x \\le \\sqrt{r \\psi_{ac}(r)},$ which follows from the Cauchy-Schwarz inequality. To see the last inequality, note that $\\psi_{ac}(r)$ goes to zero as $r$ goes to zero. \nHowever, it is easy to see that $$\\int_{0}^1 \\psi_1(x){\\rm{d}}x < \\int_0^1 \\psi(x){\\rm{d}}x .$$ Thus a priori $\\psi_{1}$ need not be an element of $\\mathcal{B}_{\\alpha}$; (see \\eqref{restr304}). To ensure that indeed $\\int_0^1\\psi_{1}(x){\\rm{d}}x \\ge \\frac{1}{2}+\\alpha$, we have to make another modification.\nNote that already in the proof of Lemma \\ref{l:uni} we argued that a priori even if $\\psi$ is not unique, $\\dot \\psi$ is unique. Thus there exist $c_{\\alpha}>0$ and $00$, denoted by $I_1$ and $I_2$, both of measure $k$; (we take the intersection with $[\\varepsilon,1]$ to ensure that both the sets are away from $0$). Note that this can always be done if $\\mu(I)> 2k$. In case $\\mu(I)\\le 2k$, we can modify $\\psi_{1}$ by taking it to be a linear function interpolating $(0,k-k_*)$ and $(r,\\psi(r))$ for some $k_*$ small enough so that $\\mu(I)>2k_*$; (see Figure \\ref{fig9} ii.)\nNothing changes in any of the arguments that follow as well as the conclusions and hence we will pretend that $k=k_*$ throughout the rest of the argument.\nLet us also choose $I_1$ and $I_2$ \nsuch that $\\sup(I_1)< \\inf(I_2)$, \nand let $\\psi_2(x):=\\int_0^x \\dot \\psi_2(y){\\rm{d}}y$ where \n\\begin{equation}\\label{modi24}\n\\dot \\psi_2:=\\dot \\psi_1 + \\frac{r}{k} \\mathbf{1}(I_1) - \\frac{r}{k}\\mathbf{1}(I_2).\n\\end{equation}\nWe will also choose $r<\\varepsilon$.\nNote that since $\\dot \\psi_1=\\dot \\psi \\ge c_1$ on $I_2$, for all small enough $r,$ $\\dot \\psi_2$ is non-negative. Also, since the measures of $I_1$ and $I_2$ are the same, $$\\int_0^1 \\dot \\psi_2(x){\\rm{d}}x =\\int_0^1 \\dot \\psi_1(x){\\rm{d}}x .$$\n\nWe now compute the area under the curve $\\psi_2(\\cdot)$ \nand see how it differs from that of $\\psi_1(\\cdot)$.\nFor every $y\\in [0,1],$ we have $$\\psi_2(y)-\\psi_1(y)=\\frac{r}{k}[ \\mu(I_1\\cap [0,y])- \\mu(I_2\\cap [0,y])].$$\nThus, \n\\begin{align}\n\\label{area123}\n\\int_0^1 [\\psi_2(y)-\\psi_1(y)]{\\rm{d}}y &=\\frac{r}{k} \\int_0^1 [\\mu(I_1\\cap [0,y])-\\mu(I_2\\cap [0,y])]{\\rm{d}}y\\\\\n\\nonumber\n&= \\frac{r}{k} \\int_{0}^1 (1-y) [\\mathbf{1}(I_1)-\\mathbf{1}(I_2)]{\\rm{d}}y\\\\\n\\nonumber\n& \\ge rk.\n\\end{align}\nThe last inequality follows since, as $\\mu(I_1)=\\mu(I_2)= k$ and $\\sup(I_1)< \\inf(I_2)$, the integral above is at least $k^2$.\nThus, the new function $\\psi_2(x)$ traps at least as much area as does $\\psi(x)$ since \n\\begin{align*}\n\\int_{0}^{1}\\psi_2(x){\\rm{d}}x -\\int_{0}^{1}\\psi(x){\\rm{d}}x &=[\\int_{0}^{1}\\psi_2(x){\\rm{d}}x -\\int_{0}^{1}\\psi_1(x){\\rm{d}}x ]+[\\int_{0}^{1}\\psi_1(x){\\rm{d}}x -\\int_{0}^{1}\\psi(x){\\rm{d}}x ],\\\\\n&\\ge rk-\\frac{rk}{2}-O(\\psi_{ac}(r)r)>0,\n\\end{align*}\n for all small enough $r$; (the last inequality uses \\eqref{arealoss}).\n\nMoreover, by Taylor expansion (using \\eqref{modi24} and $\\dot \\psi_1\\ge c_1$ on $I_1 \\cup I_2$), \n\\begin{align*}\n\\int_0^1 \\sqrt{\\dot \\psi_2(x)}{\\rm{d}}x -\\sqrt{\\dot \\psi_1(x)}{\\rm{d}}x &\\ge \\int_{I_1} \\sqrt{\\dot \\psi_1(x)}(\\frac{r}{2k\\dot \\psi_1(x)}- C(\\frac{r}{k\\dot \\psi_1(x)})^2){\\rm{d}}x \\\\\n& +\\int_{I_2} \\sqrt{\\dot \\psi_1(x)}(-\\frac{r}{2k\\dot \\psi_1(x)}- C(\\frac{r}{k\\dot \\psi_1(x)})^2){\\rm{d}}x \\\\\n\\nonumber\n&\\ge -O(\\frac{r}{k})k=-O(r).\n\\end{align*}\nWe use the bound $\\dot \\psi_1> c_1$ and that $\\mu(I_1)=\\mu(I_2)=k$ crucially in the last inequality, and all the constants depend only on $c_1$ through the Taylor expansion.\nThus, by choosing $r\\ll k$ we have $\\sqrt{rk}\\gg r$, and hence using \\eqref{lengthgain}, \n\\begin{align*}\n\\int_0^1 \\sqrt{\\dot \\psi_2(x)}{\\rm{d}}x -\\int_0^1 \\sqrt{\\dot \\psi(x)}{\\rm{d}}x &=\\int_0^1 \\sqrt{\\dot \\psi_2(x)}{\\rm{d}}x -\\int_0^1 \\sqrt{\\dot \\psi_1(x)}{\\rm{d}}x + \\int_0^1 \\sqrt{\\dot \\psi_2(x)}{\\rm{d}}x -\\int_0^1 \\sqrt{\\dot \\psi_1(x)}{\\rm{d}}x ,\\\\\n&\\ge -O(r) + \\sqrt{kr} -O(\\sqrt{\\psi_{ac}(r) r}),\\\\\n&\\ge \\sqrt{kr}\/2,\n\\end{align*}\nfor all small enough $r$. Hence, we obtain a contradiction to the extremality of $\\psi$.\n\\qed\n\n\n\\subsection{Proof of Lemma \\ref{decreasing1}}\n(i) The proof follows by taking the monotone rearrangement and using uniqueness. Let $\\dot \\psi_{mon}$ be the monotone rearrangement of $\\dot \\psi$. \nNotice that, by Fubini's theorem, \n\\begin{align}\n\\label{fubini}\n\\frac{1}{2}+\\alpha\\le \\int_0^1 \\psi(x){\\rm{d}}x =\\int_0^1 \\dot \\psi(x) (1-x){\\rm{d}}x \\le \\int_0^1 (\\dot \\psi)_{mon}(x) (1-x){\\rm{d}}x = \\int_0^1 \\psi_1(x){\\rm{d}}x \n\\end{align} \nwhere $\\dot \\psi_1=(\\dot \\psi)_{mon}$. Only the second inequality above needs justification and it is a consequence of a standard rearrangement inequality. Thus $\\psi_1$ satisfies the area constraint. \nAlso clearly $ \\int_0^1 \\sqrt {\\dot\\psi}(x){\\rm{d}}x = \\int_0^1 \\sqrt {\\dot\\psi_1}(x){\\rm{d}}x $ as rearrangement keeps integrals unchanged. \nThis contradicts the uniqueness of $\\psi$ unless $\\psi=\\psi_{1}$\n\n(ii) Let $\\dot\\psi$ be zero on a set of positive measure. By the first part of the lemma, this implies that there exists $a$ such that $\\psi(y)$ is a constant on the interval $[a,1]$. Note that it is easy to contradict extremality of $\\psi$ if $\\psi$ is not one on this interval. In case it is one on this interval, consider the function $\\psi_1(x)=1-\\psi^{-1}(1-x)$ (where $\\psi^{-1}(1)=a$). We claim that \n\\begin{align*}\n\\int_0^1 \\psi_1(y){\\rm{d}}y&=\\int_0^1 \\psi(x){\\rm{d}}x ,\\,\\,\\text{ and }\n\\int_0^1 \\sqrt{\\dot{\\psi_1}(y)}{\\rm{d}}y=\\int_0^1 \\sqrt{\\dot\\psi(x)}{\\rm{d}}x .\n\\end{align*}\nThe first equality follows by Fubini's theorem. The second becomes clear after the change of variable $y=1-\\phi(x)$ is made.\nNow, by uniqueness, $ \\psi_1=\\psi$, and by Lemma \\ref{l:ac} $\\psi$ \nhas no singular part. However, $\\psi_1$ has an atom of mass $1-a$ at $0$ which implies $a=1$.\n\\qed\n\n\n\n\\subsection{Proof of Lemma \\ref{crude1}} The reader may find it useful to refer to Figure~\\ref{fig1}. Let $\\omega_1$ and $\\omega_2$ be the acute angles that the line segments $((n,0),u)$ and $((n,0),v)$ make with the \n$x$-axis and the $y$-axis. Now, without loss of generality, we can assume $0\\le \\omega_1 \\le \\pi\/4$; otherwise, we could work with $\\omega_2$, since clearly one of the two quantities is at most $\\pi\/4$.\nNow, as a simple consequence of Theorem \\ref{lln2}, we see that, with probability at least $1-e^{-cn},$ $(n,0)$ is at a $\\Theta(n)$ distance from $\\Gamma^*_{\\alpha,n}$. Also, since $\\psi_{\\alpha}$ is strictly concave, and $\\omega_1 \\le \\pi\/4$, the line segment $(u,v)$ makes at least an angle $c(\\alpha)>0$ (depending only on $\\alpha$ and not on $u$ and $v$) with the $x$-axis. \nThe proof is now completed by considering the triangle $((n,0),u,v)$ and simple geometric arguments. We omit the details. \n\\qed\n\\subsection{Proof of Theorem \\ref{concentration1}} The proof will follow by constructing a suitable martingale and appealing to well known concentration results for martingales. However things are slightly complicated by the possibility that the martingale may not have bounded increments. A simple truncation argument will allow us to take the increments to be bounded. \nOur proof strategy follows closely arguments in \\cite{BB}.\nThe proof relies on some coarse graining. We start by introducing some notation. \nLet $$A_{i}:=\\{(x,y):i\\le x+y < {i+1}\\}.$$ \nAlso let $B_{i,j}= [{i-1}, {i})\\times [{j-1}, {j})$. Fix a number $k$ to be specified soon. Recall the point process $\\Pi$ and the path $\\Gamma=\\Gamma_{\\alpha,n}$ from Section \\ref{def10}. Let $\\Gamma_{k}=\\Gamma_{k,\\alpha,n}$ be the increasing path between $(0,0)$ and $(n,n)$ with the same constraints as $\\Gamma$ along with the additional constraint that the intersection with $A_i$ is less than $k$ for all $i \\in [0,\\ldots, 2n-1]$. Let us denote by $|\\Gamma_{k}|$ the weight of $\\Gamma_{k}$. Consider the Doob Martingale $\\{\\mathbb{E}\\left[|\\Gamma_k|\\mid \\mathcal{F}_i\\right]\\}_{0\\le i\\le 2n-1}$, \nalong the filtration $\\mathcal{F}_i=\\{\\Pi \\cap A_j : j\\le i\\}$. Note that the martingale increments are deterministically bounded by $k$.\nThe following is a standard consequence of the Azuma-Hoeffding inequality. \n\\begin{lemma} $\\mathbb{P}(\\bigl| |\\Gamma_{k}|-\\mathbb{E}(|\\Gamma_{k}|)|\\ge t)\\le e^{-\\frac{t^{2}}{4k^2n}}$.\n\\end{lemma}\n\nIn the remainder of the section, we obtain a bound on $\\bigl| |\\Gamma|-|\\Gamma_{k}|\\bigr|$ and thus also on $\\bigl|\\mathbb{E}|\\Gamma|-\\mathbb{E}|\\Gamma_{k}|\\bigr|$.\nLet $\\Pi_*$ be the point process obtained from $\\Pi$ by removing points arbitrarily if necessary from $\\Pi \\cap B_{i,j}$ to make sure that $|\\Pi_{*}\\cap B_{ij}| \\le k\/3$ for all $1\\le i,j\\le n$.\nLet $\\Gamma_{*}:=\\Gamma_{\\alpha,*}$ be the longest path with the same constraints as $\\Gamma_{\\alpha}$ in the environment $\\Pi_{*}$.\n\n\\begin{lemma} Deterministically as a consequence of the definitions it follows that $|\\Gamma_{k}| \\ge |\\Gamma_{*}|$.\n\\end{lemma}\n\n\\begin{proof} Clearly it suffices to show that $\\Gamma_{*} $ intersects none of the $A_{i}$'s at more than $k$ points. The proof is by contradiction: assume that there exists $i$ such that \n$|\\Gamma_*\\cap A_i|\\ge k+1$. Since the points on $\\Gamma_*$ are totally ordered (recall the ordering introduced in Section \\ref{def1012}), let $(x_1,y_1)$ and $(x_2,y_2)$ be the smallest and largest\npoints in the set $\\Gamma_* \\cap A_i$. By definition of $A_i$, \n\\begin{equation}\\label{last}\n(x_2-x_1)+(y_2-y_1) \\le 1,\n\\end{equation} \nso that $\\Gamma_*\\cap A_i$ intersects at most three $B_{i,j}'s$, since the number of different indices it can cross in each direction is at most one if \\eqref{last} is to be satisfied.\nThus $\\Gamma_*\\cap A_i$ \nhas to intersect at least one of the boxes at more than $\\frac{k+1}{3}$ points. This contradicts the definition of $\\Gamma_*$.\n\\end{proof}\nThus, it follows that $|\\Gamma|-|\\Gamma_{k}|\\le |\\Gamma|-|\\Gamma_*| \\le C$, where $C=\\sum_{i,j} \\max(|\\Pi_{i,j}|-k\/3,0)$ and $\\Pi_{i,j}=\\Pi \\cap B_{i,j}$. Taking $k=6\\frac{\\log n}{\\log\\log n}$, the proof is now complete in view of the next result. \n\\begin{lemma}The random variable $C$ satisfies the following:\n\\begin{enumerate} \n\\item\n$\\mathbb{P}(C\\ge \\lambda n^{1\/2}\\frac{\\log n}{\\log \\log n}) \\le 2 \\lambda ^2 e^{-\\lambda^2}$\n\\item $\\mathbb{E}(C)\\le 1$.\n\\end{enumerate}\n\\end{lemma}\n\nThis is a simple consequence of the observation that $|\\Pi_{i,j}|$ are independent Poisson variables with mean one and the following tail bound of a standard Poisson variable $X$:\n$P(X>r)\\le e^{-r\\log r +r}$.\n\\qed\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Real-Time Maude and Sockets}\n\\label{sec:maude-background}\n\nIn this section we briefly cover the important constructs we used from \nReal-Time Maude and Maude Sockets. We assume the reader is familiar with basic \nMaude constructs including modules ({\\tt mod}), sorts ({\\tt sort}), operators \n({\\tt op}), unconditional and conditional equations ({\\tt eq} and {\\tt ceq}) \nand unconditional and conditional rules ({\\tt rl} and {\\tt crl}).\n\n\\subsection{Full Maude and Real-Time Maude}\n\nFull Maude \\cite{fullmaude} is a Maude interpreter written in Maude, which in\naddition to the Core Maude constructs provides syntactic constructs such as\nobject oriented modules. Object oriented (OO) modules implicitly add sorts\n{\\tt Object} and {\\tt Msg}. Furthermore, OO-modules add a sort called {\\tt\nConfiguration} which consists of a multiset of terms of sort {\\tt Object} or {\\tt\nMsg}.\n\nObjects are represented as records:\n\n\\begin{footnotesize}\n\\begin{alltt}\n< {\\em objectID} : {\\em classID} | {\\em AttributeName} : {\\em Attribute}, ... >\n\\end{alltt}\n\\end{footnotesize}\n\nRewrite rules are then used to describe state transitions of objects\nbased on consumption of messages. For example, the following rule expresses\nthe fact that a pacemaker object consumes a message to set the pacing\nperiod to T:\n\n\\begin{footnotesize}\n\\begin{alltt}\nrl setPeriod(pm, T)\n < pm : Pacing-Module | pacing-period : PERIOD >\n => < pm : Pacing-Module | pacing-period : T > .\n\\end{alltt}\n\\end{footnotesize}\n\nReal-Time Maude \\cite{rt-maude} is a real-time extension of \nMaude in Full Maude. It adds syntactic constructs for defining timed \nmodules. Timed modules automatically import the {\\tt TIME} module, which defines \nthe sort {\\tt Time} (which can be chosen to be discrete or continuous) along\nwith various arithmetic and comparison operations on {\\tt Time}. Timed modules\nalso provide a sort {\\tt System} which encapsulates a {\\tt Configuration} and\nimplicitly associates with it a time stamp of sort {\\tt Time}. After defining a\ntime-advancing strategy, Real-Time Maude provides timed execution ({\\tt\ntrew}), timed search ({\\tt tsearch}), which performs search on\na term of sort {\\tt System} based on the time advancement strategy, and\ntimed and untimed LTL model checking commands.\n\n\\subsection{Deterministic Timed Rewriting in Real-Time Maude}\n\nWe are interested in emulations of real-time systems specified in Real-Time \nMaude. For useful real execution, a self-evident condition is that the \ntime-advancing rewrite rules in the specification should be deterministic. This \ncan be achieved by defining only one time-advancing rewrite rule on the system \nwith two auxiliary operators {\\em tick} and {\\em mte} \\cite{rtmaude-manual}. In \nour specification this is captured in {\\tt TIME-ADV-SEMANTICS}, which is \nincluded by all other timed modules:\n\n\\begin{footnotesize}\n\\begin{verbatim} \n(mod TIME-ADV-SEMANTICS is ...\n op mte : Configuration ~> TimeInf . ...\n eq mte(none) = INF .\n eq mte(C C') = minimum(mte(C), mte(C')) .\n op tick : Configuration Time ~> Configuration . ...\n eq tick(none, T) = none .\n eq tick(C C', T) = tick(C, T) tick(C', T) .\n op def-te : -> Time .\n op max-te : -> TimeInf .\n\n crl {GC} => {tick(GC, T)} in time T if T <= mte(GC) [nonexec] .\nendm)\n\\end{verbatim}\n\\end{footnotesize}\n\nThe rewrite rule at the end is assumed to be the only timed rewrite rule (a \nrewrite rule that advances the time stamp of the system) in the system \nspecification. The {\\tt mte} operator defines the maximum time elapse before \nany 0-time rewrite rule can be applied. The {\\tt tick} operator defines how \nthe system state changes due to time advancement between applications of \n0-time rewrite rules. We also define a default time elapse, {\\tt def-te}, \nand maximum time elapse, {\\tt max-te}, inside the module to be used as \nparameters during real execution.\n\n\\subsection{Socket Programming in Maude}\n\nMaude supports the Berkley sockets API for TCP communication. This is done by \nhaving a special gateway object, denoted {\\tt <>}, to consume all the messages \nresponsible for setting up sockets and communicating to an external environment \n(e.g. {\\tt createClientTcpSocket}, {\\tt send}, {\\tt receive}). The gateway \nobject will also generate messages upon status updates from the socket (e.g. \n{\\tt sent}, {\\tt received}, {\\tt closedSocket}). Consuming and generating \nmessages from the gateway object is captured by external rewrite rules which \ncan be executed using the {\\tt erew} command in Core Maude. An important thing \nworth pointing out about external rewrite rules is that {\\em external rewrite \nrules are only applied when no internal rewrite rules can be applied}. Also, \nusing external rewrite rules with Real-Time Maude specifications (built on top \nof Full Maude) requires reflecting the specification down to a Core Maude \nmodule before executing.\n\n\\section{Case Studies}\n\n\\label{sec:case}\n\n\\subsection{A Pattern for Medical Device Execution}\n\nWe briefly described in the introduction at a high level that the model \nexecution framework is to support rapid prototyping of instantiated medical \ndevice safety patterns. In \\cite{sun-meseguer-sha-wrla10} and \\cite{tech-rep}, \nwe have described in detail the command shaper pattern for medical device \nsafety. In essence, the command shaper pattern can modify commands to an \nexisting medical device to guarantee specific safety properties in terms of \nlimiting durations of stressful states and limiting the rate of change (Figure \n\\ref{fig:commandshaper}).\n\n\\begin{figure}\n\\centering \\includegraphics[width=10cm, angle=0]{figures\/pacemaker_wrapper}\n\\caption{Command Shaper Pattern for a Pacemaker}\n\\label{fig:commandshaper}\n\\end{figure}\n\n\n\\subsection{Pacemaker Simulation Case Study}\n\nOne of the applications for the command shaper pattern is a pacemaker system \n\\cite{sun-meseguer-sha-wrla10}. At a high level the safety properties \nguaranteed by the command shaper pattern is that the pacemaker will not pace at \nfast heart rates too frequently or for too long, and the pacing rate will \nchange gradually. We omit the details of instantiating the medical device \npattern, but the final wrapper object provided by the pattern is:\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(tomod PARAM-PACEMAKER is pr EPR-WRAPPER-EXEC{Safe-Pacer} ...\n eq wrapper-init =\n < pacing-module : EPR-Wrapper{Safe-Pacer} |\n inside :\n < pacing-module : Pacing-Module |\n nextPace : t(0),\n period : safe-dur >,\n val : safe-dur,\n next-val : safe-dur,\n disp : t(period),\n stress-intervals : (nil).Event-Log{Stress-Relax} > .\n... endtom)\n\\end{verbatim}\n\\end{footnotesize} \n\nThis says that a wrapper is placed around a pacing module, and the initial \npacing rate is set as the default safe-duration ({\\tt safe-dur} is 750 ms or 80 \nheart beats per minute). Verifying this instantiation (with a simple pacemaker \nlead model \\cite{tech-rep}) indicates that the safety properties are met by the \npattern. However, with the power of the model emulation framework, we can \nimmediately use this specification to run with an actual pacemaker. In this \npaper we demonstrate this emulation capability not on an actual pacemaker but \non a pacemaker simulator (a Java widget that receives messages about when to \npace and draws a simple line graph resembling an ECG trace). Before the system \ncan be emulated with the pacemaker simulator, some interface information must \nbe provided. The entire module providing all the necessary interface \ninformation is shown below:\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(mod CREATE-TICKER is\n inc PARAM-PACEMAKER .\n inc TIME-CLIENT .\n inc SEND-RECEIVE-CLIENT .\n\n eq addr = \"localhost\" .\n eq port = 4444 .\n\n eq def-te = 1 .\n eq max-te = INF .\n eq time-grain = 10 . --- milliseconds\n\n op pacer-client : -> Oid .\n eq internal = wrapper-init .\n \n eq out-adapter(shock)\n = createSendReceiveClient(pacer-client, \"localhost\", 4451, \"shock\") .\n eq in-adapter(msg-received(pacer-client, \"shocked\\n\"))\n = set-period(pacing-module, 50) .\nendm)\n\\end{verbatim}\n\\end{footnotesize} \n\nThe module first indicates that the TCP\nsocket interface to the pacemaker simulator is {\\em localhost} on {\\em port}\n4444. The default time elapse for one tick is 1 time unit. The maximum time elapse\nfor one tick step is infinity (i.e. there is no maximum). The duration of one time\nunit is 10 milliseconds. The time units are in terms of milliseconds since the\nminimum time granularity provided by the Java time interfaces is 1 millisecond.\n\nThe equation for {\\tt internal} specifies that the internal configuration to be\nexecuted is the configuration defined by {\\tt wrapper-init} (as defined in {\\tt\nPARAM-PACEMAKER}). Also, the last two equations specify that the output message\nshock should be mapped to a string ``shock'' sent over the socket, and upon\nreceiving the acknowledgment message ``shocked'' set the pacing period to 500\nms (120 bpm - a really fast heart rate). The last equation creates the scenario\nwhere a stressful heart rate is always being sent to the pacing module. Since\nthe command shaper pattern should prevent this unsafe behavior, we should see\nthe pacing automatically slow down from 120 bpm after some time interval.\n\nThe module is executed by first reflecting the {\\tt CREATE-TICKER} module down \nto Core-Maude (with the command {\\tt show all CREATE-TICKER}), and executing \nwith the {\\tt erew} command. A snapshot of the ``ECG'' trace of the pacemaker \nsimulator is show in Figure \\ref{fig:pacing}. For validation, we measured the \njitter for executing such a system -- the physical time required to completely \nexecute 0-time rules and finish communication (Figure \\ref{fig:jitter}). The \nresults were obtained from a 1.67 GHz Dual-Core Intel Centrino with Maude \nrunning in Windows through Cygwin (tracing was turned off). The main thing to \nnotice is that the jitter is mostly below 0.1 seconds and almost never exceeds \n0.2 seconds. This amount of jitter is tolerable since most medical devices need \nto respond in the order of seconds. The pacemaker is a bit more strict in terms \nof its timing requirements. To evaluate suitability for the pacemaker, we \nplotted the recorded the physical time duration between pacing events (Figure \n\\ref{fig:pacingintervals}). Notice in this example the heart rate increases \n(duration decreases) up to a limit and then the heart rate starts to decrease \n(duration increases) and the cycle repeats. It is clear that the jitter in \ncontrol seems tolerable since there are no sharp spikes in the graph of the \npacing durations.\n\n\n\\begin{figure}\n\\centering \\includegraphics[width=6cm, angle=0]{figures\/pacing}\n\\caption{Trace from Pacemaker Simulator}\n\\label{fig:pacing}\n\\end{figure}\n\n\\begin{figure}\n\\centering \\includegraphics[width=12cm, angle=0]{figures\/execution_jitter}\n\\caption{Model Execution Jitter Distribution}\n\\label{fig:jitter}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering \\includegraphics[width=12cm, angle=0]{figures\/pacing_intervals}\n\\caption{Pacing periods recorded by the pacemaker simulator (jitter effects\nare reflected by noise on the curve)}\n\\label{fig:pacingintervals}\n\\end{figure}\n\n\\subsection{Syringe Pump Case Study}\n\nThe pacemaker emulation example was demonstrated through a simulated\npacemaker mostly because current pacemakers do not have external interfaces for\nsetting when to pace (and rightly so). However, for devices such as electronic\nsyringe pumps these interfaces are available. Syringe pumps and infusion pumps\nin general deliver intravenous injections into a patient. For this scenario, we\nassume that the syringe pump is delivering an analgesic (e.g.\\ morphine) to the\npatient, and we would like to prevent overdose. We assume that for a normal\npatient overdoses do not occur at the base rate of infusion and can only result\nif a bolus dose is administered too often for the patient. We again use the\ncommand shaper pattern to limit the frequency and duration of bolus doses. The\ninstantiated pump is as follows:\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(tomod PARAM-PUMP is\n pr EPR-WRAPPER-EXEC{Safe-Pump} .\n pr DELAY-MSG .\n...\n eq msgs-init =\n delay(set-mode(pump-module, bolus), t(9))\n delay(set-mode(pump-module, bolus), t(11))\n delay(set-mode(pump-module, bolus), t(12)) ... .\n eq wrapper-init =\n < pump-module : EPR-Wrapper{Safe-Pump} |\n inside :\n < pump-module : Pump-Module |\n mode : base\n > base,\n val : base,\n next-val : base,\n disp : t(period),\n stress-intervals : (nil).Stress-Relax-Log > .\n... endtom)\n\\end{verbatim}\n\\end{footnotesize}\n\nThis module shows the initialized wrapper object for the pump, with the initial\nstate being the base rate of infusion. Furthermore, there is also a set of\ndelayed messages that will be sent to the pump. In the term {\\tt msgs-init}, the\nmodel will send bolus requests at 9 time units, 11 time units, 12 time units,\n\\ldots after the start of execution for the system. Again, creating a simulated\npatient model, we can verify the safety of the instantiated pattern\n\\cite{tech-rep}. Instantiating the pump is similar to instantiating the\npacemaker, except that there are a few more types of output messages.\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(mod CREATE-TICKER is\n inc PARAM-PUMP .\n inc TIME-CLIENT .\n inc SEND-RECEIVE-CLIENT .\n\n eq addr = \"localhost\" .\n eq port = 4444 .\n\n eq def-te = 1 .\n eq max-te = INF .\n eq time-grain = 1000 . --- milliseconds\n\n ops pump-client pump-client' : -> Oid .\n eq internal = wrapper-init msgs-init .\n \n eq out-adapter(stop)\n = createSendReceiveClient(pump-client, \"localhost\", 1234, \"STP\") .\n eq out-adapter(base)\n = createSendReceiveClient(pump-client, \"localhost\", 1234, \"RAT1\")\n createSendReceiveClient(pump-client', \"localhost\", 1234, \"RUN\") .\n eq out-adapter(bolus)\n = createSendReceiveClient(pump-client, \"localhost\", 1234, \"RAT2\")\n createSendReceiveClient(pump-client', \"localhost\", 1234, \"RUN\") .\n var S : String .\n eq in-adapter(msg-received(pump-client, S))\n = none .\n eq in-adapter(msg-received(pump-client', S))\n = none .\nendm)\n\\end{verbatim}\n\\end{footnotesize}\n\nThe model is communicating with {\\em localhost} on {\\em port} 4444. The time \ngranularity is 1 second. The internal configuration being executed is the \nwrapped pump as well as the set of messages that will deliver bolus requests. \nThe output requests are handled by a Java thread listening on port 1234 and \nforwarding the request string to the actual {\\em Multi-Phaser NE-500} Syringe \nPump (Figure \\ref{fig:pump}). A few important requests to the pump are: {\\tt \nSTP} stop the pump, {\\tt RAT } set infusion rate to {\\tt n} ml\/hr, {\\tt RUN} \nstart the infusion. Reflecting down the {\\tt CREATE-TICKER} module and \nexecuting with {\\tt erew} will now control the physical pump motor!\n\n\\begin{figure}\n\\centering \\includegraphics[width=8cm, angle=0]{figures\/pump}\n\\caption{Multi-Phaser NE-500 Syringe Pump}\n\\label{fig:pump}\n\\end{figure}\n\nAs a validation for correct pump control, we used a Salter Brecknell 7010SB \nscale to weigh the amount of liquid infused from the syringe pump over time \n(Figure \\ref{fig:pumpdata}). The data granularity is a bit rough since the \nscale can only measure within a precision of 0.1 oz. For this example, to \nclearly distinguish between two pump states, we let the base rate of infusion \nbe zero (horizontal parts of the graph) and the bolus rate be the maximum \ninfusion rate provided by the pump (positive sloped parts of the graph). Bolus \nrequests are continuously sent to the pump. The safety properties require that \nbolus doses last no longer than 30 seconds, and there must be 10 seconds \nbetween bolus doses, and at most 3 bolus doses for a window size of 3 minutes. \nThe graph validates that these properties are indeed satisfied for this \nparticular execution of the pump.\n\n\\begin{figure}\n\\centering \\includegraphics[width=8cm, angle=0]{figures\/pump_data}\n\\caption{Infusion Volume over Time}\n\\label{fig:pumpdata}\n\\end{figure}\n\n\n\n\\section{Conclusion}\n\\label{sec:conc}\n\nSafety of medical devices and of their interoperation is an unresolved issue \ncausing severe and sometimes deadly accidents for patients. Formal methods, \nparticularly in support of highly generic and reusable formal patterns whose \nsafety properties have been verified can help in ensuring the safety of \nspecific components, but this still leaves several open problems including: (i) \nhow to pass from specifications to code and from logical time to physical time \nin a correctness-preserving ways; and (ii) how to experimentally validate \nmedical safety architectures in realistic scenarios in which actual devices and \nmodels of patients and doctors can interact with formally specified and \nprovably safe designs of device components.\n\nBy developing virtual emulation environments in which highly generic and \nreusable formally verified patterns in Real-Time Maude can be easily \ntransformed into emulations in physical time which can interact with other real \ndevices and with simulations of patient and\/or doctor behaviors, we have taken \nsome first steps towards a seamless integration of formal specification and \nverification with emulation and testing, and ultimately with deployment of \nmedical DES systems that offer much stronger safety guarantees than what is \ncurrently available. Much work remains ahead. As we explain in \n\\cite{sun-meseguer-sha-wrla10}, the provably safe formal pattern used in the \nexperiments of this paper is just one such pattern: it covers a useful class, \nbut does not cover other kinds of safety needed in other medical devices. Also, \nother safety concerns, such as so-called open-loop safety, ensuring that \nmedical devices will always be in states safe for the patient even under key \ninfrastructure failures, such as network disconnection, have not been addressed\nin this work. However, we believe that the general methodology presented here \nto pass from formal specifications to virtual emulation environments and \neventually to deployed systems should also be applicable to those new formally \nverified patterns that have yet to be developed.\n\n\n\\section{Introduction}\n\n\\label{sec:intro}\n\nEach year, just in the US hospitals, a shocking and almost unacceptable number \nof medical accidents occur. In a 2009 study, reports estimate 40,000 instances \nof medical harm occur daily, and from the 2005 through 2007 period, at least \n92,882 deaths were potentially preventable \\cite{HealthGrades2009}. Many of \nthese accidents happen due to mistakes and failures in the \n\\emph{interoperation} of medical devices. A modern hospital's operating room is \nin fact a quite complex distributed embedded system (DES) with many devices \ninvolved in either passively monitoring the patient state or actively \nperforming different parts of a procedure. Both the safety of the individual \ndevices and the safety of interoperation between devices (and between the \npatients and doctors) are of paramount importance. Presently, this safety is \nnot adequately guaranteed.\n\nWhen the reported accidents are analyzed, it becomes clear that many of them \ncould and should have been avoided if the DES formed by the devices, the \npatient, and the doctors had been properly designed and analyzed, so that many \nunsafe interactions become \\emph{impossible} by design. The use of formal \nmethods can clearly help in this respect, and promising research advances have \nalready been made in this direction (see, e.g., \n\\cite{alur04,arney07,ray04,jetley06}).\n\nIn our recent work \\cite{tech-rep}\\cite{sun-meseguer-sha-wrla10}, we have \ndeveloped a scalable and highly reusable approach to the safety of medical \ndevices by means of \\emph{formally verified patterns} that: (i) are formally \nspecified as real-time rewrite theories in Real-Time Maude; (ii) are generic, \nso that they apply not to a single device but to a wide range of devices, and \nare therefore specified as \\emph{parameterized} modules; (iii) come with \nexplicit \\emph{formal requirements} (specified in their parameter theories) \nthat must be met by any pattern instantiation to be correct; and (iv) come with \n\\emph{formal safety guarantees} that will be satisfied by any correct \ninstantiation of the pattern. For example, in \\cite{tech-rep} we present one \nsuch pattern and show how it can be instantiated to obtain safe controllers for \nquite different devices, such as a pacemaker, an infusion pump for analgesia, \nand the interoperation of a ventilator with an X-ray machine.\n\nHowever, there is still a substantial gap between the verified safety of \ndesigns in formal specifications and the actual safety of real medical devices, \npatients and doctors in an operating room for at least two reasons. First, the \nformal specifications are somewhat idealized abstractions in which, for \nexample, time is not the actual physical time that devices need to operate in, \nbut logical time, and the code of the actual devices and controllers is not \nused but instead some formal specifications are used. Second, it is important \nto consider not just the safety of a single device or small collection of \ndevices, but also that of their \\emph{interoperation} with other devices and \nwith the patient and the doctors. This work takes some first steps towards the \ngoal of bridging the gap between formal specifications and actual devices in a \nhospital to help ensure that safety properties are preserved in the passage \nfrom specification to actual code and physical devices. To achieve this goal\nwe propose the use of \\emph{virtual emulation environments} in which:\n\\begin{enumerate}\n\\item formally verified patterns \\cite{sun-meseguer-sha-wrla10} can be \ninstantiated to obtain various concrete specifications of desired devices and \ncontrollers;\n\n\\item the so-obtained formal executable specifications of devices and \ncontrollers are used \\emph{directly} to generate emulators that perform the \nsame specified behavior in physical time;\n\n\\item actual devices, as well as actual executable models of patient and doctor \nbehavior, can be seamlessly integrated with specification-based emulators to \nvalidate the safety not just of individual devices but also of various \nDESs that are needed in practice in actual operating room conditions and \nscenarios.\n\\end{enumerate}\nThe advantage of point (1) is a great degree of reusability, and amortizing the \nformal verification effort across potentially many devices. The advantage of \n(2) is that, since each specification-based emulator executes the exact same \nformal specification that has been proved safe for the given device, the safety \nof such an emulator is automatically guaranteed, and a path remains open to \ncorrectly generate actual code for it preserving such safety in an actual \nimplementation. The advantage of point (3) is that system experimentation with \nphysical time and actual devices becomes available from very early in the \ndesign process and are available afterwards along the entire development \nprocess: initially, only specifications may be emulated; at intermediate \nstages, both specifications and actual devices form the virtual emulation \nenvironment; and in the end the emulating environment seamlessly becomes an \nactual implementation.\n\nTechnically, the way such virtual emulation environments are obtained from \nformal specifications is by using a key idea first demonstrated by Musab \nAl-Turki to semi-automatically pass from a Real-Time Maude formal executable \nspecification operating in \\emph{logical time} to a corresponding \n\\emph{physical emulation} of the same specification operating in \\emph{physical \ntime} and possibly interacting with other devices in a distributed way. The \nkey observations are: (i) in Real-Time Maude rewrite rules are either 0-time \nrules requiring no time, or time-advancing rules moving the entire system \nforward in logical time; (ii) time advancing rules (typically a single such \nrule) can be physically implemented by an external object that sends time ticks \naccording to physical time; (iii) although so-called 0-time rules do take some \nphysical time to be executed, if this time is small enough in comparison with \nthe time granularity of the physical time period chosen, for all practical \npurposes they can be considered to take 0 time units to execute; and (iv) the \nMaude infrastructure for Maude computations to interact with external objects \nvia sockets can be used to interface the Maude objects in the formal \nspecification with the external ticker object and also to other external \ndevices.\n\n\\subsection{Our Contribution.} This is the first work we are aware of in which \nformal specifications of real-time components are directly used in the area of \ndistributed embedded systems for medical applications to obtain a virtual \nemulation environment in which specifications, patients, doctors, and actual \ndevices can be emulated in physical time, and such that the correctness of \nverified specifications is preserved, provided adequate timing \nconstraints are obeyed (see Section \\ref{sec:issues}). This work is a first \nstep towards a seamless integration of formal specification, verification, and \nsystem development and testing for safe medical systems. The associated notion \nof a virtual emulation environment plays a crucial role in passing from \nspecifications to code and devices, and from logical time to physical time. We \nhave demonstrated both the feasibility and the usefulness of these methods in \ntwo concrete scenarios: one in which a pacemaker interacts adaptively in \nphysical time with a simulated model of a patient heart and keeps heart rates \nwithin a safe envelope; and another in which a safety controller for \npatient-controlled analgesia interacts in physical time with an actual drug \ninfusion device and with a simulation of patient behavior.\n\nIn the passage from formal real-time specifications to their corresponding \nemulators there are additional novel contributions that were required for this \nwork, including: (i) advancing time by the maximum time elapsable as opposed to \nby a fixed ticking period; (ii) handling asynchronous interrupts in addition to \nsynchronous communication; (iii) emulating the interaction of real medical \ndevices, patient models, and formal specifications; and (iv) generating a timed \nwrapper for each component specification almost for free with deterministic \nReal-Time Maude specifications using both a time ticker and the computation of \nthe maximum time elapsable for each time advance.\n\nThe paper is organized as follows: Section \\ref{sec:overview} provides the high \nlevel ideas of the framework from formal patterns to real-time execution. \nSection \\ref{sec:maude-background} covers the basics of Real-Time Maude and \nMaude's support for socket programming. Section \\ref{sec:io} and \n\\ref{sec:emulation} describes the core of the execution framework which allows \nseamless passage from Real-Time Maude specifications to execution with physical \ntime and physical devices. Section \\ref{sec:case} covers case studies for a \npacemaker and a syringe pump to evaluate the feasibility of using formal models \nto execute medical devices. Finally, we describe some fundamental assumptions \nrequired for formal model execution to work for medical devices in Section \n\\ref{sec:issues}, and we conclude in Section \\ref{sec:conc}.\n\\section{Mapping Internal Messages to External I\/O}\n\\label{sec:io}\n\nValidating the design of a device in an execution environment requires handling \nits outputs. After all, the end validation of a system's behavior is based on \nits outputs. Thus, it seems reasonable to talk about how internal messages in \nthe model can be converted into messages for communicating with the external \nworld. This section also serves as an explanation for unfamiliar readers of how \nMaude sockets are used.\n\nIn order to talk about external communication, we must first define in the \nmodel what is external. The model will have an internal distributed actor \nconfiguration with internal messages as well as messages to be output to the \nexternal world. Thus, the first definition is {\\tt EXTERNAL-CONFIGURATION} \nwhich defines external messages {\\tt ExtMsg} as subsort of {\\tt Msg}. \nFurthermore, external messages are classified in terms of incoming external \nmessages {\\tt InExtMsg} and outgoing external messages {\\tt OutExtMsg}. A \nconfiguration is called {\\em open} if there are external messages present in \nthe configuration: either an incoming external message has not been delivered, \nor an outgoing external message has not been sent. The predicate {\\tt open?} is \ndefined accordingly.\n\n\\begin{footnotesize}\n\\begin{verbatim} \nsubsorts InExtMsg OutExtMsg < ExtMsg < Msg .\nop open? : Configuration -> Bool .\n eq open?(C C') = open?(C) or open?(C') .\n eq open?(O) = false .\n eq open?(M) = M :: ExtMsg .\n\\end{verbatim}\n\\end{footnotesize}\n\nActually sending an external message may be more complex than just forwarding \nthe message through the gateway object. External messages may not be the same \nin the internal configuration and in the external configuration. For example, a \nsimple output message in the internal configuration may need to be mapped to a \nclient object that initiates the communication to deliver the message. \nOperators {\\tt in-adapter} and {\\tt out-adapter} are defined to perform these \nmappings from external message client configurations to internal messages.\n\nAn example of an output adapter for a pacemaker message to beat the heart\nmay be:\n\n\\begin{footnotesize}\n\\begin{verbatim}\neq out-adapter(shock)\n = createSendReceiveClient(pacer-client, \"localhost\", 4451, \"SetLeadVoltage 5V\")\n\\end{verbatim}\n\\end{footnotesize}\n\nIn this example, the message {\\tt shock} is transformed into a client object \nwhich sends a message on port 4451 with the string {\\tt \"SetLeadVoltage 5V\"} \nindicating that the proxy server will then proceed to set a 5V voltage on the \npacemaker lead.\n\n\\subsection{One-Round Communication Clients}\n\nOnce the external message is mapped into a client configuration, we must define\nthe rewrite rules to specify how the communication protocol works with the\nexternal device. Here we describe a simple {\\tt SEND-RECEIVE-CLIENT} which is\nresponsible for establishing communication, sending a message, receiving a\nreply, and then closing the communication. Although simple, this type of\nprotocol is sufficient for most of the communication for medical devices we have\nused in our case studies.\n\n\\begin{footnotesize}\n\\begin{verbatim}\n(mod SEND-RECEIVE-CLIENT is ...\n op createSendReceiveClient : Oid String Nat String -> Configuration .\n eq createSendReceiveClient(CLIENT, ADDRESS, PORT, SEND-CONTENTS)\n = < CLIENT : SendReceiveClient | ... >\n createClientTcpSocket(socketManager, CLIENT, ADDRESS, PORT) .\n op msg-received : Oid String -> InExtMsg .\n...endm)\n\\end{verbatim}\n\\end{footnotesize}\n\nAfter creating the client and establishing communication, the client goes into \none round of send and receive before the socket is closed. Once the socket is \nclosed, the entire client object is converted into one reply message to be \ndelivered to the internal configuration using the operator {\\tt msg-received}.\n\n\\begin{footnotesize}\n\\begin{verbatim}\n--- send contents\nrl createdSocket(CLIENT, socketManager, SOCKET-DST)\n < CLIENT : SendReceiveClient | ... send-contents : SEND-CONTENTS >\n => < CLIENT : SendReceiveClient | ... > send(SOCKET-DST, CLIENT, SEND-CONTENTS) .\n--- receive contents\nrl sent(CLIENT, SOCKET-DST) < CLIENT : SendReceiveClient | ... >\n => < CLIENT : SendReceiveClient | ... > receive(SOCKET-DST, CLIENT) .\n--- close socket\nrl received(CLIENT, SOCKET-DST, RECEIVE-CONTENTS) < CLIENT : SendReceiveClient | ... >\n => < CLIENT : SendReceiveClient | ... recv-contents : RECEIVE-CONTENTS >\n closeSocket(SOCKET-DST, CLIENT) .\n--- done\nrl closedSocket(CLIENT, SOCKET-DST, \"\")\n < CLIENT : SendReceiveClient | ... recv-contents : RECEIVE-CONTENTS >\n => msg-received(CLIENT, RECEIVE-CONTENTS) .\n\\end{verbatim}\n\\end{footnotesize}\n\n\\section{Assumptions and Issues}\n\n\\label{sec:issues}\n\nIn this section we discuss the timing assumptions that need to be taken into\naccount to ensure that the emulation of a Real-Time Maude specification\ncorrectly implements the logical time behavior.\n\nFigure \\ref{fig:timing} shows one round of communication between the time server\nand the formal model.\n$t_{comm,i}$ denotes the delays due to each stage of communication.\n$t_{rew,i}$ denotes the delays incurred by each stage of rewriting. $t_{proc}$ \ndenotes the time needed at the physical device interface to process\nthe commands. Thus, the entire time to finish a round is $t_{round} =\nt_{comm1} + t_{rew1} + t_{comm2} + t_{proc} + t_{comm3} +\nt_{rew2} + t_{comm4}$.\n\n\\begin{figure}\n\\centering \\includegraphics[width=10cm, angle=0]{figures\/timing}\n\\caption{Timing Considerations}\n\\label{fig:timing}\n\\end{figure}\n\nIn logical time, no time advancement should actually take place in a \ncommunication round. All computation and message communication is assumed to \ntake zero time. Of course, for proper timed operation we can relax these \nconstraints to first allow the model to have a non-zero (but bounded) delay for \nthese computations and communications. The maximum bound on these \ncommunications is $t_{round} \\leq mte(C_{next})$. Otherwise, by the time the \nround has completed, the execution is already delayed passed the time for the \nmaximum time elapse for the next 0-timed rewrite rule; i.e., the maximum speed \nof execution of the formal model and communication is slower than the real time \nrequirements. Now, assuming that the constraint $t_{round} \\leq mte(C_{next})$ \nis satisfied, there is still another problem we must deal with. The actual \ncommands sent to the device are not received until $t_{jitter} = t_{comm1} + \nt_{rew1} + t_{comm2}$ time after they are actually supposed to be executed. This \ncould be very problematic. Even if the model can keep up with real-time, the \ntime in which it issues commands will be delayed. For example, the shocks from \na pacemaker may be issued at the correct time by the model, but the real shock \nis not delivered until 0.1 seconds later. To meet this requirement, we need to \nlook at the finer requirements of medical devices and patient parameters. How \nmuch jitter in control can a patient tolerate? As we have seen in the Section \n\\ref{sec:case}, the jitter seems to be tolerable for the applications we \nconsidered, and furthermore, the end-to-end round communication timing \nconstraints are also satisfied by our case studies.\n\n\\section{Overview of the Model Execution Framework}\n\n\\label{sec:overview}\n\nAn envisioned design framework from generic design patterns to executable \nspecifications is shown in Figure \\ref{fig:pattern-to-execution}. We start with \na safety pattern, which is a parameteric module with well-defined parameters \nwith formal requirements provided by an input theory. The next stage is to \ninstantiate the pattern to a concrete instance, which will of course still \nsatisfy all the safety properties ensured by the pattern. Finally, in the last \nstage (the focus of this paper), the entire executable specification of a \nsystem can really be executed in the real world by encapsulating the \nspecification in an external wrapper for model execution. Figure \n\\ref{fig:pattern-to-execution} illustrates these various stages for the design \nof a cardiac pacemaker system.\n\n\\begin{figure}\n\\centering \\includegraphics[width=9cm, angle=0]{figures\/mde_process}\n\\caption{From Formal Patterns to Real-Time Execution}\n\\label{fig:pattern-to-execution}\n\\end{figure}\n\nThe first step formally defines a safety pattern as a parameterized module. In \nour pacemaker example, the pattern is a generic safety wrapper for medical \ndevices. We briefly describe the pattern in this paper, summarizing the details \npresented in \\cite{tech-rep}\\cite{sun-meseguer-sha-wrla10}. The safety wrapper \nfilters the input commands, so that state changes in a medical device fall \nwithin safe physiological ranges and constraints. The white boxes in the \ndiagram denote pattern parameters that must be instantiated. For the device \nsafety wrapper these parameters include the type of device that is being \nconsidered, the period for updating device states, states of the device \nconsidered stressful for the patient, etc. Aside from the parameters, there are \nalso formal constraints that these parameters must conform to in order for a \nparameter instance to be acceptable. For example, stress states and relaxed \nstates must be disjoint for a device.\n\nThe next stage is to instantiate the parameters of the pattern. In our \nexample, the medical device safety wrapper is instantiated to filter the input \ncommands to a pacemaker. The state of the pacemaker is assumed to be its pacing \nrate. The necessary parameters are then filled in. The period for updating the \npacing rate is every 10 ms; pacing rates between 90 bpm to 120 bpm are \nconsidered stressful for the patient; etc. This instantiated model will satisfy \nthe safety properties guaranteed by the safety pattern, provided all the \nparameters satisfy the necessary constraints and formal parameter requirements. \nFurthermore, once instantiated, the wrapped pacemaker is just another \nexecutable Real-Time Maude model. Thus, we can use this model for simulation and\nmodel checking purposes in Real-Time Maude.\n\nIn the final stage, which is the main focus of this paper, we transform the \nmodel to execute in {\\em real world} time with physical devices in a medical \ndevice emulation environment. For this purpose, we take the model of the \nwrapped pacemaker and wrap it again in an external execution wrapper (Figure \n\\ref{fig:mdewrapper}). The execution wrapper is responsible for conveying to \nthe model the notion of real world time as well as providing a communication \ninterface to the external world. A dedicated timer thread is responsible for \n``ticking'' the model by sending a minimal number of messages to advance the \nmodel's logical time. The timer thread also intercepts all asynchronous \n(interrupt) messages and relays them to the model. Another aspect of the \nexecution wrapper is the ability to map external I\/O messages to communicate \nwith the external devices. For example, in the pacemaker specification, an \ninternal message called {\\em paceVentricle} may be mapped into an entire client \nconfiguration to send a message for setting the final voltage on a pacing lead.\n\n\\begin{figure}\n\\centering \\includegraphics[width=6cm, angle=-90]{figures\/mde_wrapper}\n\\caption{Real-Time Model Execution Wrapper}\n\\label{fig:mdewrapper}\n\\end{figure}\n\nFor the design of a medical device or, more generally, of any safety-critical \nsystem, all the stages can work together to achieve a modular component design, \nand to support experimentation and testing in the context of other real devices \nthat the safe component being designed has to interact with. In the first \nstages of safety pattern specification, we create a parameteric formal \nspecification that essentially isolates important safety properties of a device \nfrom the rest of the system. We then can use theorem proving techniques to \nprovide provable properties of the safety pattern. Although theorem proving \nmay be time-consuming, as we show with our case studies one safety pattern can \nbe applied to many different applications, so the time spent proving properties \nof the pattern is well worth the effort. The second stage with pattern \ninstantiation is necessary to obtain a fully specified and executable model. \nThe last stage of course takes the existing models, with minimal auxiliary \ninformation for external interfaces, and provides an executable prototype \nessentially for free. In this way, it becomes possible to emulate the behavior\nof safe medical components in an experimental environment involving interactions\nwith real medical devices.\n\n\\section{Distributed Emulation of Safe Medical Devices}\n\\label{sec:emulation}\n\nThe external execution wrapper is an object that encapsulates the original \nformal model. It is primarily responsible for interfacing constructs between \nthe physical world (the real interfaces to devices) and the logical world (the \nworld as seen by the formal model). In particular, the execution wrapper is \nresponsible for conveying the measurement of real time elapsed to the model and \nalso for mapping logical communication messages to communication configurations \nthat can deliver the message to real devices. The most important feature of the \nexternal execution wrapper is its modularity. Aside from adding the minimal \ninformation about how to map external I\/O to messages in the model, no further \nspecifications are required to execute the logical model within an external \nenvironment.\n\n\\subsection{Mapping Logical Time to Physical Time}\n\nAs mentioned earlier, time advancement of the system is achieved by defining \nthe {\\em tick} and {\\em mte} operators. Ideally the system continuously evolves \nover time (possibly nondeterministically). Of course, we cannot capture the \nnotion of continuous time without abstractions in the model, so to advance time \ndiscretely, an {\\em mte} (maximum time elapsable) operator is introduced. A \ncorrectly defined {\\em mte} operator ensures that if a system is in state $S$, \nthen for any time $T < mte(S)$, no 0-time rewrite rules (state transitions)\n can apply to $tick(S, T)$. That is, if a system is in state $S$, and $T \\leq \nmte(S)$, then $tick(S, T)$ will be equivalent to the state $S$ advancing in \ncontinuous time for $T$ time units. This ideal semantics of time is shown on the \nleft side of Figure \\ref{fig:time}. The figure shows that 0-time rewrite \nrules are assumed to take zero time, and ideally, the system continuously \nevolves over time between the 0-time rewrite rules.\n\n\\begin{figure}\n\\centering \\includegraphics[width=14cm, angle=0]{figures\/logical_time_to_physical_time}\n\\caption{From Ideal Time Advancement Semantics to Physical Time Advancement}\n\\label{fig:time}\n\\end{figure}\n\nOf course, in a real execution of the model, the ideal notion of time with \n0-time rewrite rules and time-advancing rules is only an idealized \nabstraction. Performing rewrites cannot take zero time, and we cannot \ncontinuously rewrite states of the system over time. We could of course create \na model in discrete time with very fine time granularity and drive it by a high \nfrequency clock like in hardware. However, this would introduce a lot of \nunnecessary overhead in terms of communication of timing messages and \nperforming rewrite rules to change the model for every clock tick. We resolve \nthis problem by observing that the actual internal state of the model is not \nimportant at most instants in time unless it is communicating with the external \nworld. The model states only generate output messages with 0-time rewrite \nrules, so we can essentially let the model in state $S$ remain unaffected by \nthe passage of time until the next time instant in which a 0-time rewrite \nrule can be applied; this is exactly $mte(S)$ time units later. This method of \ndriving execution is shown in the right part of Figure \\ref{fig:time}. We have \ncreated a dedicated timer server thread (in Java) that has access to the system \ntime. When the execution of the wrapped model starts, it will send a start \nrequest which includes the time units of the model or the minimum granularity \nof time for model execution in milliseconds. Once the timer thread processes \nall the initial information, it will send a {\\em Go!} message to signal the \nmodel to start executing. The model then calculates the maximum time elapsable \n(which is 10 seconds in the example) and sends this information to the timer \nthread. The model then proceeds to sleep until the timer thread wakes it in \ntime for the next 0-time rewrite rule. The process then continues. There are \ntwo key points to notice about this example. The input and output messages from \nthe model may be delayed by an amount of time equal to the communication jitter \nplus the time to complete rewriting. Normally this delay is on the order of 10 \nms, but this is still suitable for medical devices which normally receive \ncommands on the order of seconds or more. Also, the timer thread sets the \ntimeout from the last time it sent a time advancement message to the model and \nnot from the time it receives the $mte$ message from the model. This ensures \nthat clock skew and jitter are bounded over time.\n\n\\subsection{Synchronous Timed Execution}\n\nThe {\\em communication wrapper} ({\\tt commwrap}) is represented as an object \nwith the attributes for the communication state, the socket information for \ncommunication, and the internal wrapped (Real-Time Maude) system model being \nexecuted. The top level system is of sort {\\tt CommWrapConfiguration} for any \ncommunicating model.\n\n\\begin{footnotesize}\n\\begin{verbatim}\nop commwrap : Configuration -> CommWrapConfiguration .\nop wrap-client : Configuration -> Configuration .\n eq wrap-client(C) = < client : TickClient |\n state : start, internal : [ {C} in time 0 ], socket-name : no-oid > .\nop init-client : -> CommWrapConfiguration .\n eq init-client = commwrap( <> wrap-client(internal)\n createClientTcpSocket(socketManager, client, addr, port) ) .\n\\end{verbatim}\n\\end{footnotesize}\n\nThe communication wrapper initializes a wrapped communication client that \nreceives messages from the tick server (a Java thread executing in real-time \nthat sends it messages for time advancement). After creating the TCP socket, \nthe first message sent from the client to the tick server is the \ntime-granularity ({\\tt time-grain}), which is a rational number specifying the \nnumber of milliseconds in one time unit. Then, the actual execution starts when \nthe communication wrapper receives a {\\tt GO} message from the tick server. The \ntime when the tick server sends the {\\tt GO} message is the starting point from \nwhich time elapses are being measured. Upon receiving the {\\tt GO} message, the \nformal model will immediately start to execute ({\\tt state : run}).\n\n\n\\begin{footnotesize}\n\\begin{verbatim} \nrl [send-init] :\n commwrap( <> createdSocket(...) < client : TickClient | ... > )\n => commwrap( <> < client : TickClient | ... > send(..., string(time-grain)) ) .\nrl [wait-for-go] :\n commwrap( <> sent(...) < client : TickClient | ... > )\n => commwrap( <> < client : TickClient | ... > receive(...) ) .\nrl [start-running] :\n commwrap( <> received(..., \"GO\\r\\n\") < client : TickClient | ... > )\n => commwrap( <> < client : TickClient | state : run, ... > .\n\\end{verbatim}\n\\end{footnotesize}\n\nThe formal model executes until {\\tt mte} becomes non-zero (no other 0-time \nrewrite rules can be applied), and the model sends a message to request the \nnext time advancement message after the maximum time elapse and blocks. After \nsending this waiting duration, the tick server will sleep for this time \nduration and then send a time advancement message when the time has expired. \nThe model will then advance time (tick) the model for the time duration expired\nand perform 0-time rewrite rules. The model now blocks again for the next {\\tt\nmte}, and the cycle repeats.\n\n\\begin{footnotesize}\n\\begin{verbatim} \ncrl [request-wait-timer] :\n commwrap( <>\n < client : TickClient |\n state : run,\n internal : [ {C} in time T ], ... > )\n => commwrap( <>\n < client : TickClient |\n state : request, ... >\n send(..., string(mte(C, T))) )\n if mte(C,T) :: TimeInf \/\\ mte(C,T) > 0 \/\\ not open?(C) . ...\n\nrl [block] :\n commwrap( <> sent(...)\n < client : TickClient |\n state : request, ... > )\n => commwrap( <>\n < client : TickClient |\n state : wait, ... >\n receive(SOCKET-NAME, client) ) .\n\nrl [wake-up] :\n commwrap( <> received(..., ADV-STR)\n < client : TickClient |\n state : wait, ... > )\n => commwrap( <>\n < client : TickClient |\n state : run,\n internal : [ {tick(C, rat(ADV-STR))} in time rat(ADV-STR) in time T ], ... > ) .\n\\end{verbatim}\n\\end{footnotesize}\n\n\n\\subsection{Handling Asynchronous External Events}\n\nSo far, the model can only handle synchronous events (polling and blocking \ncommunication). However, in general a useful design must be able to react to \nexternal events from the environment. For example, an EKG sensor detects a QRS \nwaveform, and sends this information to the pacemaker. This points to the fact \nthat our model needs to be able to handle external events asynchronously.\n\n\\begin{figure}\n\\centering \\includegraphics[width=8cm, angle=0]{figures\/interrupts}\n\\caption{Handling Interrupts and Asynchronous Communication Semantics}\n\\label{fig:async}\n\\end{figure}\n\nAn external message would trigger a 0-time rewrite rule to receive the message \nby some object and process it. More precisely, if we have $C_M$ and $C_{Ext}$ \nas the model configuration and the external (environment) configuration \nrespectively, the maximal time elapse for the system $C_M C_{Ext}$ should be \n$min(mte(C_M), mte(C_{Ext}))$, where $mte(C_{Ext})$ denotes time duration \nbefore the next interrupt message. This semantics is captured by having \ninterrupt messages forwarded by the timer thread, as shown in Figure \n\\ref{fig:async}. The timer thread will only check for interrupts when it is \nwaiting for the next timeout, so when the interrupt message arrives, it will \nwake up and immediately forward the interrupt message to the model with the \namount of time that has elapsed. Any future timeouts are canceled. Introducing \nthe notion of interrupts requires us to modify the wake-up rule for the model \nto not only advance time, but also check for potential interrupt messages as \nwell.\n\n\\begin{footnotesize}\n\\begin{verbatim}\nrl [wake-up] :\ncommwrap( <> received(client, SOCKET-NAME, INTR-STR)\n < client : TickClient |\n state : wait,\n internal : [ {C} in time T ],\n socket-name : SOCKET-NAME > )\n=>\ncommwrap( <> < client : TickClient |\n state : run,\n internal : [\n {tick(C, recv->rat(INTR-STR)) recv->conf(INTR-STR)}\n in time recv->rat(INTR-STR) in time T\n ],\n socket-name : SOCKET-NAME > ) .\n\\end{verbatim}\n\\end{footnotesize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Omitted Proof from \\Cref{app:universal}}\nHere we provide the proof for \\Cref{lem:bcard}.\n\\begin{proof}[Proof for \\Cref{lem:bcard}] Fix an ordering on the elements of $\\mathrm{F}$. Let $\\mathrm{F}(i)$ denote the $i$'th frequency element of $\\mathrm{F}$. For all $\\phi_{S} \\in B$, the set of distinct frequencies in $\\phi_{S}$ is a subset of $\\mathrm{F}$ and the length of $\\phi_{S}$ is equal to $n$. Therefore, any element $\\phi_{S} \\in B$ can be encoded as a unique vector $v_{\\phi_{S}} \\in [0,n]^{\\mathrm{F}}$, where $v_{\\phi_{S}}(i) \\eqdef \\phi(\\mathrm{F}(i))$ denotes the number of elements in $\\phi_{S}$ that have frequency $F(i)$. Using the previous discussion, we have $|B| \\leq |[0,n]^{\\mathrm{F}}| \\leq (n+1)^{|\\mathrm{F}|}$.\n\\end{proof}\n\n\n\\section{Omitted Proofs from \\Cref{sec:appl}}\\label{sec:omitted}\nHere, we present and prove results related to the existence of an estimator for entropy and distance to uniformity on a fixed subset $S\\subseteq \\mathcal{D}$. Note the estimator we provide here is exactly same to the one presented in~\\cite{ADOS16} but defined only on subset $S\\subseteq \\mathcal{D}$. All the results and proofs presented here are similar to the ones in~\\cite{ADOS16} and for completeness and verification purposes we reprove (with slight modifications) these results. As in~\\cite{ADOS16}, we first provide a general definition of an estimator that works both for entropy and distance to uniformity. In \\Cref{lem:general_est}, we prove a result that captures the maximum change of this general estimator by changing one sample. In section \\ref{subsec:entr} and \\ref{subsec:dtu}, we provide proofs for entropy and distance to uniformity respectively.\n\nGiven $2n$ samples $x^{2n}=(x_1^n,x_2^n)$ from distribution $\\textbf{p}$. Let\n$n_{\\smb}^{'}\\eqdef \\textbf{f}(x_1^n,y)$, and $n_{\\smb}\\eqdef \\textbf{f}(x_2^n,y)$ be the number of appearances of symbol $y$ in the first and second half respectively. We define the following estimator which is exactly the same as \\cite{ADOS16} but defined only on subset $S$. For all $x \\in S$,\n\\[\n\\hat{g}_{S}(x^{2n})\n=\\max\\left\\{\\min\\left\\{\\sum_{y \\in S}{g_{y}},f_{S,\\max}\\right\\}, 0\\right\\}.\n\\]\nwhere $f_{S,\\max}$ is the maximum value of the property $f$ on subset $S$ and for all $y \\in S$,\n\\[\ng_{y}=\n\\begin{cases}\nG_{L,g}(n_{\\smb}), & \\text{ for } n_{\\smb}^{'} 0$. Suppose $c_1=2c_2$, and $c_2>35$,\n Further suppose that\n $n^3\\left(\\frac{16c_1}{\\alpha^2}+\\frac1{c_2}\\right)>\\log k\\cdot\\log\n n$. Then for all subset $S\\subseteq \\mathcal{D}$, there exists a polynomial approximation of $- y \\log y$ {with degree $L = 0.25 \\alpha \\log n$}, over\n $[0, c_1\\frac{\\log N}{n}]$ such that $\\max_{i} |b_i| \\leq\n n^{\\alpha}\/n$ and the bias of the entropy estimator on subset $S$ ($\\sum_{y \\in S}\\textbf{p}_{y} \\log \\frac{1}{\\textbf{p}_{y}}$) is at most\n $O\\left(\\left(1+\\frac{1}{\\alpha^2}\\right)\\frac{N}{n\\log N} + \\frac{\\log N}{N^4}\\right)$.\n\\end{lemma}\n\\begin{proof}\n\t\\newcommand{{E}_{1}}{{E}_{1}}\n\t\\newcommand{{E}_{2}}{{E}_{2}}\n\t\\newcommand{{E}_{3}}{{E}_{3}}\n\t\\newcommand{{E}}{{E}}\n\t\\newcommand{n_{y}}{n_{y}}\n\t\\newcommand{n'_{y}}{n'_{y}}\n\t\\newcommand{\\expt}[1]{\\mathbb{E}\\left[ #1 \\right]}\n\t\nWe first upper bound the bias of our estimator. We consider three events, \n$${E}_{1}\\eqdef \\cap _{y \\in S} \\left\\{ n'_{y} \\leq c_{2} \\log N, n'_{y} \\leq c_{1} \\log N \\implies \\textbf{p}_{y} \\leq \\frac{c_1 \\log N}{n}\\right\\},$$ \n$${E}_{2}\\eqdef\\cap _{y \\in S} \\left\\{ n'_{y} > c_{2} \\log N \\implies \\textbf{p}_{y} > \\frac{c_3 \\log N}{n}\\right\\},$$ $${E}_{3}\\eqdef \\cap _{y \\in S}\\left\\{ n'_{y} \\leq c_{2} \\log N \\text{ and } n'_{y} > c_{1} \\log N \\right\\}~.$$ \nBy proof of Lemma 6 in~\\cite{ADOS16} we have,\n\\begin{equation}\\label{prob:e3}\n\\Prob{{E}_{3}^c} \\leq \\frac{1}{n^{4.9}}\n\\end{equation}\nBy equations 48 and 49 in~\\cite{WY16} combined with \\Cref{prob:e3}, we get,\n$$\\Prob{{E}_{1}^c} \\leq \\frac{2}{N^4} \\text{ and } \\Prob{{E}_{2}^c} \\leq \\frac{1}{N^4} $$\nDefine ${E}\\eqdef {E}_{1} \\cap {E}_{2}$, then \n\\begin{equation}\\label{prob:eall}\n\\Prob{{E}^c} \\leq \\Prob{{E}_{1}^c}+\\Prob{{E}_{2}^c} \\leq \\frac{2}{N^4}~. \n\\end{equation}\n\n\\newcommand{I_1}{I_1}\n\\newcommand{I_2}{I_2}\nFurther we define random sets\n$I_1\\eqdef \\left\\{y \\in S | n'_{y} < c_{2} \\log N, n_{y} < c_{1} \\log N \\text{ and } \\textbf{p}_{y} \\leq \\frac{c_1 \\log N}{n}\\right\\}$ and $I_2 \\eqdef \\left\\{ y \\in S ~|~n'_{y} > c_{2} \\log N \\text{ and } \\textbf{p}_{y} > \\frac{c_3 \\log N}{n}\\right\\}$.\nWe first bound the conditional bias and we later use it to bound the bias of our estimator. Our next statement follows from uniform approximation error~\\cite{Tim63} and is explicitly written in to Equation 53 of~\\cite{WY16}.\n\\begin{equation}\\label{eq:2}\n|\\expt{\\textbf{f}_{I_1}(\\textbf{p})-\\hat{g}_{I_1} | I_1}=|\\sum_{y \\in I_1} \\textbf{p}_{y} \\log \\frac{1}{\\textbf{p}_y}-P_{L,g}(\\textbf{p}_{y})| \\leq \\frac{N}{\\alpha^2 n \\log N}~.\n\\end{equation}\nLet $\\hat{\\textbf{p}}$ be the empirical distribution on $x_2^n$. Similarly by analysis of Case 2 and Equation 58 in~\\cite{WY16} we have,\n\\begin{equation}\\label{eq:3}\n|\\expt{\\textbf{f}_{I_2}(\\textbf{p})-\\hat{g}_{I_2} | I_2}=|\\expt{\\sum_{y \\in I_2} (\\textbf{p}_{y} \\log \\frac{1}{\\textbf{p}_y}-\\hat{\\textbf{p}}_{y} \\log \\frac{1}{\\hat{\\textbf{p}}_y})|I_2}| \\leq \\frac{N}{n \\log N}~.\n\\end{equation}\nCombining equations \\ref{eq:2}, \\ref{eq:3}, \\ref{prob:eall} and \\ref{prob:e3} we can upper bound the bias of our estimator by $\\frac{2N}{n \\log N}+\\frac{4 \\log N}{N^4}$. Note here we use the fact that in the case of bad event (${E}^c$ or ${E}_{3}^c$) the bias of our estimator is upper bounded by $\\log N$.\n\nOur analysis for largest change in the value of estimator by changing one sample is exactly the same as~\\cite{ADOS16} and for completeness we describe it next. The largest coefficient of the optimal uniform polynomial approximation of degree $L$ for function $x \\log x$ in the interval $[0,1]$ is upper bounded by $2^{3L}$. This result follows from the proof of Lemma 2 in~\\cite{CL11} and is also explicitly mentioned in the proof of Proposition 4 in~\\cite{WY16}. Therefore, the largest change (after appropriately normalizing) is the largest value of $b_i$ (co-efficient of the optimal uniform polynomial approximation) which is\n\\[\n\\frac{2^{3L}e^{L^2\/n}}{n}.\n\\]\nFor $L = 0.25\\alpha\\log n$, this is at most $\\frac {n^\\alpha}{n}$. \n\\end{proof}\n\nThe proof of Lemma~\\ref{thm:entrochange} for entropy follows from the above lemma and Lemma~\\ref{lem:general_est} by substituting $n= O\\left(\\frac{N}{\\log N} \\frac{1}{\\epsilon}\\right)$ and $\\epsilon>\\Omega\\left(\\frac{\\log N}{N^{1-\\alpha}}\\right)$.\n\n\\subsection{Distance to uniformity}\\label{subsec:dtu}\nHere we provide proof sketch for the existence of an estimator with desired properties. This result is analogous to Lemma 7 in~\\cite{ADOS16} and proof for this lemma is very similar to that of \\cite{ADOS16} and for completeness we sketch the proof for this result. \n\\begin{lemma}\n\n Let $c_1>2c_2$, $c_2=35$. Then for all subset $S\\subseteq \\mathcal{D}$, there is an estimator for distance to uniformity on subset $S$ ($\\sum_{y \\in S}|\\textbf{p}_{y}-\\frac{1}{N}|$) that changes by at most $n^{\\alpha}\/n$ when a sample is changed, and the bias of the estimator is at most $O(\\frac{1}{\\alpha}\\sqrt{\\frac{c_1\\log N}{N\\cdot n}})$.\n\\end{lemma}\n\\begin{proof}\nWe divide estimation of distance to uniformity into two cases based on $n$. Note the proof for this lemma follows along the lines of \\cite{ADOS16}.\n\\paragraph{Case 1: $\\frac1N < c_2\\log N\/n$.} In this case, we use the estimator defined in the last section for $g(x) = |x-1\/k|$.\n\n\\paragraph{Case 2: $\\frac1N > c_2\\log N\/n$.} The estimator is as follows for all $y \\in S$:\n\\[\ng_{y}=\n\\begin{cases}\nG_{L,g}(n_{\\smb}), & \\text{ for }\n\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_2\\log N}{Nn}}, \\text{\n and } \\absv{\\frac{n_{\\smb}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_1\\log N}{Nn}},\\\\ 0,\n& \\text{ for }\n\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}<\\sqrt{\\frac{c_2\\log N}{Nn}}, \\text{\n and } \\absv{\\frac{n_{\\smb}}{n}-\\frac1{N}}\\ge\\sqrt{\\frac{c_1\\log N}{Nn}},\\\\ g\n\\left(\\frac{n_{\\smb}}{n}\\right) , & \\text{ for }\n\\absv{\\frac{n_{\\smb}^{'}}{n}-\\frac1{N}}\\ge\\sqrt{\\frac{c_2\\log N}{Nn}}.\n\\end{cases}\n\\]\n\nThe estimator proposed in~\\cite{ADOS16} is exactly the same as ours, but we define our estimator only for the domain elements in $S\\subseteq \\mathcal{D}$. Note the estimator defined in~\\cite{JHW17} is slightly different, assigning $G_{L,g}(n_{\\smb})$ for the first two cases. As in~\\cite{ADOS16}, this second case is designed to bound the change in value of the estimator by changing one sample. Using~\\cite{Tim63}(Equation 7.2.2),~\\cite{JHW17} (for their estimator) show that, contribution towards bias (conditioned on \"good\" event \\footnote{\\label{event}Refer~\\cite{JHW17} for definitions of \"good\" and \"bad\" events.}) by any domain element $y \\in \\mathcal{D}$ (note we only need this result to hold for $y\\in S$) satisfying $n_{\\smb}^{'}\\frac{1}{k^{0.2499}}$. Further for any $\\epsilon<\\frac{1}{k^{\\delta}}$ for some constant $\\delta>0$, the empirical distribution based plug-in estimator is exact with high probability. Here we provide proofs for two main results described in \\Cref{sec:results} for support. In \\Cref{thm:supp} and \\Cref{thm:approxsupp}, we show that PML and approximate PML distributions (under the constraint that all its probability values are $\\geq \\frac{1}{k}$) based plug-in estimators are sample complexity optimal for all parameter regimes, thus providing a better analysis for \\cite{ADOS16}.\n\nWe next define a function that outputs the number of distinct frequencies in the profile. Later in \\Cref{lem:pmlprob}, we show that the support of PML and approximate PML distribution is at least the number of distinct elements in the sequence.\n\\begin{defn}\n\tFor any $S \\subseteq \\mathcal{D}$, the function $\\mathrm{Distinct}:\\Phi^{n}\\rightarrow \\mathbb{Z}_{+}$, takes input $\\phi$ and returns $\\sum_{j\\in[n]}\\phi_j$. For any sequence $x^n$, we overload notation and use $\\mathrm{Distinct}(x^n)$ to denote $\\mathrm{Distinct}(\\Phi(x^n))$. Note $\\mathrm{Distinct}(\\phi)$ and $\\mathrm{Distinct}(x^n)$ denote the number of distinct domain elements observed in profile $\\phi$ or sequence $x^{n}$ respectively.\n\t\\end{defn}\n\\begin{lemma}\\label{lem:pmlprob}\n\tFor any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$ such that $\\textbf{p}_x \\in \\{0\\} \\cup [\\frac{1}{k},1]$ and a profile $\\phi \\in \\Phi^{n}$, if $S(\\textbf{p})<\\mathrm{Distinct}(\\phi)$ then $\\textbf{p}(\\phi)=0$.\n\\end{lemma}\n\\begin{proof}\nConsider sequences $x^{n}$ with $\\Phi(x^n)=\\phi$. All such sequences have $\\mathrm{Distinct}(\\phi)$ number of distinct observed elements that is strictly greater than $S(\\textbf{p})$ and distribution $\\textbf{p}$ assigns probability zero for all these sequences.\n\\end{proof}\n\\newcommand{\\mathrm{Support}}{\\mathrm{Support}}\n\\begin{proof}[Proof for \\Cref{thm:supp}]\n\tGiven $\\phi$, let $\\textbf{p}_{\\phi} \\in \\Delta^{\\mathcal{D}}$, be the distribution with $\\textbf{p}_{\\phi}(x) \\in \\{0\\}\\cup[\\frac{1}{k},1]$. If $\\epsilon>\\frac{1}{k^{0.2499}}$, we know that plug-in approach on $\\textbf{p}_{\\phi}$ is sample complexity optimal~\\cite{ADOS16}. We consider the regime where $\\epsilon \\leq \\frac{1}{k^{0.2499}}$ and here the number of samples $n=c \\cdot k\\log k$ for some constant $c \\geq 2$. If $S(\\textbf{p}_{\\phi}) < \\mathrm{Distinct}(\\phi)$, then by \\Cref{lem:pmlprob} we have $\\textbf{p}_{\\phi}(\\phi)=0$ a contradiction because the empirical distribution assigns a non-zero probability value for observing $\\phi$. Therefore, without loss of generality we assume $S(\\textbf{p}_{\\phi}) \\geq \\mathrm{Distinct}(\\phi)$. We next argue that $S(\\textbf{p}_{\\phi}) = \\mathrm{Distinct}(\\phi)$. We prove this statement by contradiction. Suppose $S(\\textbf{p}_{\\phi}) > \\mathrm{Distinct}(\\phi)$, then define $\\textbf{p}_{\\phi}' \\in \\Delta^{\\mathcal{D}}$ to be the PML distribution under constraints $S(\\textbf{p}_{\\phi}')=\\mathrm{Distinct}(\\phi)$ and $\\textbf{p}_{\\phi}'(x) \\in \\{0\\}\\cup[\\frac{1}{k},1]$. Let $\\mathrm{Support}:\\Delta^{\\mathcal{D}} \\rightarrow 2^{\\mathcal{D}}$ be a function that takes distribution $\\textbf{p}$ as input and returns index set for the support of $\\textbf{p}$. Now consider $\\Prob{\\textbf{p}_{\\phi},\\phi}$, and recall $\\Prob{\\textbf{p}_{\\phi},\\phi}=\\sum_{\\{x^n\\in \\mathcal{D}^n~|~\\Phi(x^n)=\\phi\\}}\\Prob{\\textbf{p}_{\\phi},x^n}$. Further note that $\\sum_{\\{x^n\\in \\mathcal{D}^n~|~\\Phi(x^n)=\\phi\\}}\\Prob{\\textbf{p}_{\\phi},x^n}=\\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\sum_{\\{x^n\\in S^n~|~\\Phi(x^n)=\\phi\\}}\\Prob{\\textbf{p}_{\\phi},x^n}$, therefore,\n\\begin{equation}\\label{eq:a1}\n\\begin{split}\n\\Prob{\\textbf{p}_{\\phi},\\phi}&=\\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\sum_{\\{x^n\\in S^n~|~\\Phi(x^n)=\\phi\\}}\\Prob{\\textbf{p}_{\\phi},x^n}\\\\\n& \\leq \\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\left(1-\\frac{S(\\textbf{p}_{\\phi})-|S|}{k} \\right)^n \\Prob{\\textbf{p}_{\\phi}',\\phi}~.\n\\end{split}\n\\end{equation}\n\\newcommand{\\textbf{p}_{\\phi,S}}{\\textbf{p}_{\\phi,S}}\nIn the second inequality, we use for all $x \\in \\mathcal{D}$, $\\textbf{p}_{\\phi}(x)\\in \\{0\\}\\cup [\\frac{1}{k},1]$ and we have $\\sum_{x\\in S}\\textbf{p}_{\\phi}(x) \\leq \\left(1-\\frac{S(\\textbf{p}_{\\phi})-|S|}{k} \\right)$ and the inequality follows. \n\nWe next upper bound the term $\\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\left(1-\\frac{S(\\textbf{p}_{\\phi})-|S|}{k} \\right)^n$. Note that, \n\\begin{align*}\n\\sum_{\\{S\\subseteq \\mathrm{Support}(\\textbf{p}_{\\phi})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} & \\left(1-\\frac{S(\\textbf{p}_{\\phi})-|S|}{k} \\right)^n \\leq \\expo{-n\\frac{S(\\textbf{p}_{\\phi})-\\mathrm{Distinct}(\\phi)}{k}} \\binom{S(\\textbf{p}_{\\phi})}{\\mathrm{Distinct}(\\phi)} \\\\\n& \\leq \\expo{-n\\frac{S(\\textbf{p}_{\\phi})-\\mathrm{Distinct}(\\phi)}{k} + (S(\\textbf{p}_{\\phi})-\\mathrm{Distinct}(\\phi))\\log S(\\textbf{p}_{\\phi}) }\\\\\n& \\leq \\expo{ -\\log k }~.\n\\end{align*}\nIn the second inequality, we use a weak upper bound on the quantity $\\binom{S(\\textbf{p}_{\\phi})}{\\mathrm{Distinct}(\\phi)}$. In the third and fourth inequality, we use $n=ck\\log k$, $c\\geq2$ and $k \\geq S(\\textbf{p}_{\\phi}) > \\mathrm{Distinct}(\\phi)$. Combining everything together we get, $\\Prob{\\textbf{p}_{\\phi},\\phi} \\leq \\expo{ -\\log k }\\Prob{\\textbf{p}_{\\phi}',\\phi}$. A contradiction because $\\textbf{p}_{\\phi}$ is the PML distribution.\n\n\n\nTherefore if $n>2k\\log k$, then the previous derivation implies, \n$$\\Prob{S(\\textbf{p}_{\\phi})=\\mathrm{Distinct}(\\phi)}=1~.$$ \nFurther if $n>2k\\log k$, then \n$$\\Prob{S(\\textbf{p}) =\\mathrm{Distinct}(\\phi)} \\geq 1-k\\expo{\\frac{-n}{k}}~.$$ \nCombining previous two inequalities and substituting $n>2k\\log k$ we get, $\\Prob{S(\\textbf{p}) =S(\\textbf{p}_{\\phi})} \\geq 1-\\exps{-\\log k}$, thus concluding the proof.\n\\end{proof}\n\n\\renewcommand{\\bp^{\\beta}_{\\phi_{S}}}{\\textbf{p}^{\\beta}_{\\phi}}\n\\begin{proof}[Proof for \\Cref{thm:approxsupp}]\n\t\tThe proof for this result is similar to \\Cref{thm:supp} and for completeness we reprove it. Given $\\phi$, let $\\textbf{p}_{\\phi},\\bp^{\\beta}_{\\phi_{S}} \\in \\Delta^{\\mathcal{D}}$, be PML and $\\beta$-approximate PML distributions respectively under the constraint $\\textbf{p}_{\\phi}(x),\\bp^{\\beta}_{\\phi_{S}}(x) \\in \\{0\\}\\cup[\\frac{1}{k},1]$. If $\\epsilon>\\frac{1}{k^{0.2499}}$, by \\cite{ADOS16} we already know that plug-in approach on $\\bp^{\\beta}_{\\phi_{S}}$ for $\\beta=\\exps{-\\epsilon^2 n^{1-\\alpha}}$ is sample complexity optimal with high probability. Here we consider the regime $\\epsilon \\leq \\frac{1}{k^{0.2499}}$ and in this case the number of samples $n=c \\cdot k\\log k$ for some large constant $c \\geq 2$. If $S(\\bp^{\\beta}_{\\phi_{S}}) < \\mathrm{Distinct}(\\phi)$, then by \\Cref{lem:pmlprob} we have $\\bp^{\\beta}_{\\phi_{S}}(\\phi)=0$ which is a contradiction, because the empirical distribution clearly returns a non-zero probability value. Therefore, without loss of generality we assume $S(\\bp^{\\beta}_{\\phi_{S}}) \\geq \\mathrm{Distinct}(\\phi)$. We next argue that $S(\\bp^{\\beta}_{\\phi_{S}}) \\leq \\mathrm{Distinct}(\\phi)+\\epsilon k$. We prove this statement by contradiction. Suppose $S(\\bp^{\\beta}_{\\phi_{S}}) > \\mathrm{Distinct}(\\phi)+\\epsilon k$, then consider the $\\Prob{\\bp^{\\beta}_{\\phi_{S}},\\phi}$, and recall $\\Prob{\\bp^{\\beta}_{\\phi_{S}},\\phi}=\\sum_{\\{x^n\\in \\mathcal{D}^n|\\Phi(x^n)=\\phi\\}}\\Prob{\\bp^{\\beta}_{\\phi_{S}},x^n}$. Further note that $\\sum_{\\{x^n\\in \\mathcal{D}^n|\\Phi(x^n)=\\phi\\}}\\Prob{\\bp^{\\beta}_{\\phi_{S}},x^n}=\\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})||S|=\\mathrm{Distinct}(\\phi) \\}} \\sum_{\\{x^n\\in S^n|\\Phi(x^n)=\\phi\\}}\\Prob{\\bp^{\\beta}_{\\phi_{S}},x^n}$. Therefore,\n\t\\begin{equation}\n\t\\begin{split}\n\t\\Prob{\\bp^{\\beta}_{\\phi_{S}},\\phi}&=\\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})||S|=\\mathrm{Distinct}(\\phi) \\}} \\sum_{\\{x^n\\in S^n|\\Phi(x^n)=\\phi\\}}\\Prob{\\bp^{\\beta}_{\\phi_{S}},x^n}\\\\\n\t& \\leq \\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\left(1-\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-|S|}{k} \\right)^n \\Prob{\\textbf{p}_{\\phi},\\phi}~.\n\t\\end{split}\n\t\\end{equation}\n\tIn the final inequality we used for all $x \\in \\mathcal{D}$, $\\bp^{\\beta}_{\\phi_{S}}(x)\\in \\{0\\}\\cup [\\frac{1}{k},1]$ and we have $\\sum_{x\\in S}\\bp^{\\beta}_{\\phi_{S}}(x) \\leq \\left(1-\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-|S|}{k} \\right)$ and using the definition of $\\textbf{p}_{\\phi}$ the inequality follows. \n\t\n\tWe next upper bound the term $\\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} \\left(1-\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-|S|}{k} \\right)^n$. Note that, \n\t\\begin{align*}\n\t\\sum_{\\{S\\subseteq \\mathrm{Support}(\\bp^{\\beta}_{\\phi_{S}})~|~|S|=\\mathrm{Distinct}(\\phi) \\}} & (1-\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-|S|}{k} )^n \\leq \\exps{-n\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi)}{k}} \\binom{S(\\bp^{\\beta}_{\\phi_{S}})}{\\mathrm{Distinct}(\\phi)} \\\\\n\t& \\leq \\exps{-n\\frac{S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi)}{k} + (S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi))\\log S(\\bp^{\\beta}_{\\phi_{S}}) } \\\\\n\t& \\leq \\exps{ (S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi))(\\log k- c \\log k } \\\\\n\t& \\leq \\exps{ -\\epsilon k \\log k } < \\exps{ -\\epsilon^2 n^{1-4\\alpha} }~.\n\t\\end{align*}\n\t\\newcommand{\\textbf{p}_{\\phi,S}}{\\textbf{p}_{\\phi,S}}\n\t In the second inequality, we use a weak upper bound for the quantity $\\binom{S(\\textbf{p}_{\\phi})}{\\mathrm{Distinct}(\\phi)}$. In the third and fourth inequality, we use $n=ck\\log k$, $c\\geq2$ and $S(\\bp^{\\beta}_{\\phi_{S}}) > \\mathrm{Distinct}(\\phi)+\\epsilon k$. In the final inequality, we use $n^{1-\\alpha}\\leq k\\log k$ for constant $\\alpha>0$. Combining everything together we get $\\Prob{\\bp^{\\beta}_{\\phi_{S}},\\phi} < \\exps{ -\\epsilon^2 n^{1-4\\alpha} } \\Prob{\\textbf{p}_{\\phi},\\phi}$, a contradiction on the definition of $\\bp^{\\beta}_{\\phi_{S}}$.\n\t\n\tTherefore if $n>2k\\log k$, then the previous derivation implies $\\Prob{|S(\\bp^{\\beta}_{\\phi_{S}})-\\mathrm{Distinct}(\\phi)|\\geq \\epsilon k}=1$. Further if $n>2k\\log k$, then $\\Prob{S(\\textbf{p}) =\\mathrm{Distinct}(\\phi)} \\geq 1-k\\exps{\\frac{-n}{k}}$. Combining the previous two inequalities and substituting $n>2k\\log k$ we get, $\\Prob{|S(\\bp^{\\beta}_{\\phi_{S}})-S(\\textbf{p})|\\geq \\epsilon k} \\geq 1-\\exps{-\\log k}$, thus concluding the proof.\n\t\\end{proof}\n\\section{Experiments}\\label{sec:exp}\nWe performed two different sets of experiments for entropy estimation -- one to compare performance guarantees and the other to compare running times. In our pseudo PML approach, we divide the samples into two parts. We run the empirical estimate on one (this is easy) and the PML estimate on the other. For the PML estimate, any algorithm to compute an approximate PML distribution can be used in a black box fashion. \nAn advantage of the pseudo PML approach is that it can use any algorithm to estimate the PML distribution as a black box, providing both competitive performance and running time efficiency. \nIn our experiments, we use the heuristic algorithm in \\cite{PJW17} to compute an approximate PML distribution. In the first set of experiments detailed below, we compare the performance of the pseudo PML approach with raw \\cite{PJW17} and other state-of-the-art estimators for estimating entropy. Our code is available at \\url{https:\/\/github.com\/shiragur\/CodeForPseudoPML.git}\n\\begin{figure}[!ht]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{plotJuly30cite.pdf}\n\t\\label{fig:entr}\\end{figure}\n\\vspace{-24pt}\n\nEach plot depicts the performance of various algorithms for estimating entropy of different distributions with domain size $N=10^5$. \nEach data point represents 50 random trials. ``Mix 2 Uniforms'' is a mixture of two uniform distributions, with half the probability mass on\nthe first $N\/10$ symbols, and $\\mathrm{Zipf}(\\alpha) \\sim 1\/i^{\\alpha}$ with $i \\in [N]$. \nMLE is the naive approach of using the empirical distribution with correction bias;\nall the remaining algorithms are denoted using bibliographic citations.\nIn our algorithm we pick $threshold=18$ (same as \\cite{WY16}) and our set $\\mathrm{F}=[0,18]$ (input of \\Cref{algpml}), i.e. we use the PML estimate on frequencies $\\leq 18$ and empirical estimate on the rest. Unlike \\Cref{algpml}, we do not perform sample splitting in the experiments -- we believe this requirement is an artifact of our analysis. For estimating entropy, the error achieved by our estimator is competitive with \\cite{PJW17} and other state-of-the-art entropy estimators. Note that our results match \\cite{PJW17} for small sample sizes because not many domain elements cross the threshold and for a large fraction of the samples, we simply run the \\cite{PJW17} algorithm.\n\n\nIn the second set of experiments we demonstrate the running time efficiency of our approach. In these experiments, we compare the running time of our algorithm using \\cite{PJW17} as a subroutine to the raw \\cite{PJW17} algorithm on the $\\mathrm{Zipf(1)}$ distribution. The second row is the fraction of samples on which our algorithm uses the empirical estimate (plus correction bias). The third row is the ratio of the running time of \\cite{PJW17} to our algorithm. \nFor large sample sizes, the entries in the EmpFrac row have high value, i.e. our algorithm applies the simple empirical estimate on large fraction of samples; therefore, enabling $10$x speedup in the running times. \n\\vspace*{-0.1in}\n\\begin{center}\n\t\\begin{tabular}{|c||c|c|c|c|c|c|c|c|} \n\t\t\\hline\n\t\tSamples size & $10^3$ & $5*10^3$ & $10^4$ & $5*10^4$ & $10^5$ & $5*10^5$ &$10^6$ & $5*10^6$ \\\\ [0.5ex] \n\t\t\\hline\n\t\tEmpFrac & 0.184 & 0.317 & 0.372 & 0.505 & 0.562 & 0.695 & 0.752 & 0.886 \\\\ \n\t\t\\hline\n\t\tSpeedup & 0.824 & 1.205 & 1.669 & 3.561 & 4.852 & 9.552 & 13.337 & 12.196 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{center}\n\\section{Introduction}\nSymmetric property estimation is a fundamental and well studied problem in machine learning and statistics. In this problem, we are given $n$ i.i.d samples from an unknown distribution\\footnote{Throughout the paper, distribution refers to discrete distribution.} $\\textbf{p}$ and asked to estimate $\\textbf{f}(\\textbf{p})$, where $\\textbf{f}$ is a symmetric property (i.e. it does not depend on the labels of the symbols).\nOver the past few years, the computational and sample complexities for estimating many symmetric properties have been extensively studied.\nEstimators with optimal sample complexities have been obtained for several properties including entropy~\\cite{VV11a, WY16, JVHW15}, distance to uniformity~\\cite{VV11b, JHW16}, and support~\\cite{VV11a, WY15}. \n\nAll aforementioned estimators were property specific and therefore, a natural question is to design a universal estimator. \nIn \\cite{ADOS16}, the authors showed that the distribution that maximizes the profile likelihood, i.e. the likelihood of the multiset of frequencies of elements in the sample, referred to as \\emph{profile maximum likelihood (PML) distribution}, can be used as a universal plug-in estimator. \\cite{ADOS16} showed that computing the symmetric property on the PML distribution is sample complexity optimal in estimating support, support coverage, entropy and distance to uniformity within accuracy $\\epsilon > \\frac{1}{n^{0.2499}}$. Further, this also holds for distributions that approximately optimize the PML objective, where the approximation factor affects the desired accuracy.\n\nAcharya et al. \\cite{ADOS16} posed two important and natural open questions. The first was to give an efficient algorithm for finding an approximate PML distribution, which was recently resolved in \\cite{CSS19}. \nThe second open question is whether PML is sample competitive in all regimes of the accuracy parameter $\\epsilon$? In this work, we make progress towards resolving this open question.\n\nFirst, we show that the PML distribution based plug-in estimator achieves optimal sample complexity for all $\\epsilon$ for the problem of estimating support size.\nNext, we introduce a variation of the PML distribution that we call the \\emph{pseudo PML distribution}.\nUsing this, we give a general framework for estimating a symmetric property.\nFor entropy and distance to uniformity, this pseudo PML based framework achieves optimal sample complexity for a broader regime of the accuracy parameter than was known for the vanilla PML distribution.\n\nWe provide a general framework that could, in principle be applied to estimate any separable symmetric property $\\textbf{f}$, meaning $\\textbf{f}(\\textbf{p})$ can be written in the form of $\\sum_{x \\in \\mathcal{D}}\\textbf{f}(\\textbf{p}_x)$. This motivation behind this framework is that for any symmetric property $\\textbf{f}$ that is separable, the estimate for $\\textbf{f}(\\textbf{p})$ can be split into two parts: $\\textbf{f}(\\textbf{p})=\\sum_{x \\in B}\\textbf{f}(\\textbf{p}_x)+\\sum_{x \\in G}\\textbf{f}(\\textbf{p}_x)$, where $B$ and $G$ are a (property dependent) disjoint partition of the domain $\\mathcal{D}$. \nWe refer to $G$ as the good set and $B$ as the bad set. Intuitively, $G$ is the subset of domain elements whose contribution to $\\textbf{f}(\\textbf{p})$ is easy to estimate, i.e a simple estimator such as empirical estimate (with correction bias) works. \nFor many symmetric properties, finding an appropriate partition of the domain is often easy.\nMany estimators in the literature~\\cite{JVHW15, JHW16, WY16} make such a distinction between domain elements.\nThe more interesting and difficult case is estimating the contribution of the bad set: $\\sum_{x \\in B}\\textbf{f}(\\textbf{p}_x)$.\nMuch of the work in these estimators is dedicated towards estimating this contribution\nusing sophisticated techniques such as polynomial approximation.\nOur work gives a unified approach to estimating the contribution of the bad set. We propose a PML based estimator for estimating $\\sum_{x \\in B}\\textbf{f}(\\textbf{p}_x)$. \nWe show that computing the PML distribution only on the set $B$ is sample competitive for entropy and distance to uniformity for almost all interesting parameter regimes thus (partially) handling the open problem proposed in \\cite{ADOS16}. \nAdditionally, requiring that the PML distribution be computed on a subset $B \\subseteq \\mathcal{D}$ reduces the input size for the PML subroutine and results in practical algorithms (See \\Cref{sec:exp}).\n\nTo summarize, the main contributions of our work are:\n\\vspace*{-0.1in}\n\\begin{itemize}\n\\item We make progress on an open problem of \\cite{ADOS16} on broadening the range of error parameter $\\epsilon$ that one can obtain for universal symmetric property estimation via PML.\n\\item We give a general framework for applying PML to new symmetric properties.\n\\item As a byproduct of our framework, we obtain more practical algorithms that invoke PML on smaller inputs (See \\Cref{sec:exp}).\n\\end{itemize}\n\n\n\\subsection{Related Work}\nFor many natural properties, there has been extensive work on designing efficient estimators both with respect to computational time and sample complexity \\cite{HJW17, HJM17, AOST14, RVZ17, ZVVKCSLSDM16, WY16a, RRSS07, WY15, OSW16, VV11a, WY16, JVHW15, JHW16, VV11b}.\nWe define and state the optimal sample complexity for estimating support, entropy and distance to uniformity. For entropy, we also discuss the regime in which the empirical distribution is sample optimal.\n\n\\noindent{\\bf Entropy:} For any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the entropy $H(\\textbf{p})\\eqdef -\\sum_{x\\in \\mathcal{D}}\\textbf{p}_{x} \\log \\textbf{p}_{x}$. For $\\epsilon \\geq \\frac{\\log N}{N}$ (the interesting regime), where $N\\eqdef |\\mathcal{D}|$, the optimal sample complexity for estimating $H(\\textbf{p})$ within additive accuracy $\\epsilon$ is $O(\\frac{N}{\\log N}\\frac{1}{\\epsilon})$~\\cite{WY16}. Further if $\\epsilon < \\frac{\\log N}{N}$, then \\cite{WY16} showed that empirical distribution is optimal.\n\n\\noindent{\\bf Distance to uniformity:} For any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the distance to uniformity $\\|\\textbf{p}-u\\|_1\\eqdef \\sum_{x\\in \\mathcal{D}}|\\textbf{p}_{x}-\\frac{1}{N}|$, where $u$ is the uniform distribution over $\\mathcal{D}$. The optimal sample complexity for estimating $\\|\\textbf{p}-u\\|_1$ within additive accuracy $\\epsilon$ is $O(\\frac{N}{\\log N}\\frac{1}{\\epsilon^2})$~\\cite{VV11b, JHW16}.\n\n\\newcommand{k}{k}\n\\noindent{\\bf Support:} For any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the support of distribution $S(\\textbf{p})\\eqdef |\\{x\\in \\mathcal{D}~|~\\textbf{p}_{x}>0 \\}|$. Estimating support is difficult in general because we need sufficiently large number of samples to observe elements with small probability values. Suppose for all $x \\in \\mathcal{D}$, if $\\textbf{p}_{x} \\in \\{0\\} \\cup [\\frac{1}{k},1]$, then \\cite{WY15} showed that the optimal sample complexity for estimating support within additive accuracy $\\epsilon k$ is $O(\\frac{k}{\\log k}\\log^2 \\frac{1}{\\epsilon})$.\n\nPML was introduced by Orlitsky et al.~\\cite{OSSVZ04} in 2004. \nThe connection between PML and universal estimators was first studied in~\\cite{ADOS16}. \nAs discussed in the introduction, PML based plug-in estimator applies to a restricted regime of error parameter $\\epsilon$.\nThere have been several other approaches for designing universal estimators for symmetric properties. Valiant and Valiant~\\cite{VV11a} adopted and rigorously analyzed a linear programming based approach for universal estimators proposed by \\cite{ET76} and showed that it is sample complexity optimal in the constant error regime for estimating certain symmetric properties (namely, entropy and support size). Recent work of Han et al.~\\cite{HJW18} applied a local moment matching based approach in designing efficient universal symmetric property estimators for a single distribution. \\cite{HJW18} achieves the optimal sample complexity in restricted error regimes for estimating the power sum function, support and entropy.\n\nRecently, \\cite{YOSW18} gave a different unified approach to property estimation.\nThey devised an estimator that uses $n$ samples and achieves the performance attained by the empirical estimator with $n \\sqrt{\\log n}$ samples for a wide class of properties and for all underlying distributions. \nThis result is further strengthened to $n \\log n$ samples for Shannon entropy and a broad class of other properties including $\\ell_1$-distance in \\cite{HO19b}.\n\nIndependently of our work, authors in \\cite{HO19a} propose \\emph{truncated PML} that is slightly different but similar in the spirit to our idea of pseudo PML. They use the approach of truncated PML and study its application to symmetric properties such as: entropy, support and coverage; refer \\cite{HO19a} for further details.\n\\subsection{Organization of the Paper}\nIn \\Cref{sec:prelims} we provide basic notation and definitions. \nWe present our general framework in \\Cref{sec:results} and state all our main results. \nIn \\Cref{app:universal}, we provide proofs of the main results of our general framework. \nIn \\Cref{sec:appl}, we use these results to establish the sample complexity of our estimator in the case of entropy (See \\Cref{sec:entr}) and distance to uniformity (See \\Cref{sec:dtu}). Due to space constraints, many proofs are deferred to the appendix. In \\Cref{sec:exp}, we provide experimental results for estimating entropy using pseudo PML and other state-of-the-art estimators. Here we also demonstrate the practicality of our approach.\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection*{Acknowledgments}\nWe thank the reviewers for the helpful comments, great suggestions, and positive feedback.\nMoses Charikar was supported by a Simons Investigator Award, a Google Faculty Research Award and an Amazon Research Award.\nAaron Sidford was partially supported by NSF CAREER Award CCF-1844855.\n\\bibliographystyle{alpha}\n\n\\section{Preliminaries}\\label{sec:prelims}\nLet $[a]$ denote all integers in the interval $[1,a]$. Let $\\Delta^{\\mathcal{D}} \\subset [0,1]_{\\mathbb{R}}^{\\mathcal{D}}$ be the set of all distributions supported on domain $\\mathcal{D}$ and let $N$ be the size of the domain. Throughout this paper we restrict our attention to discrete distributions and assume that we receive a sequence of $n$ independent samples from an underlying distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$. Let $\\mathcal{D} ^n$ be the set of all length $n$ sequences and $y^n \\in \\mathcal{D}^n$ be one such sequence with $y^n_{i}$ denoting its $i$th element. The probability of observing sequence $y^n$ is:\n$$\\mathbb{P}(\\textbf{p},y^n) \\eqdef \\prod_{x \\in \\mathcal{D}}\\textbf{p}_x^{\\textbf{f}(y^n,x)}$$\nwhere $\\textbf{f}(y^n,x)= |\\{i\\in [n] ~ | ~ y^n_i = x\\}|$ is the frequency\/multiplicity of symbol $x$ in sequence $y^n$ and $\\textbf{p}_x$ is the probability of domain element $x\\in \\mathcal{D}$. We next formally define profile, PML distribution and approximate PML distribution.\n\n\\begin{defn}[Profile]\nFor a sequence $y^n \\in \\mathcal{D}^n$, its \\emph{profile} denoted $\\phi=\\Phi(y^n) \\in \\mathbb{Z}_{+}^{n}$ is $\\phi\\eqdef(\\phi(j))_{j \\in [n]} $ where $\\phi(j)\\eqdef|\\{x\\in \\mathcal{D} |\\textbf{f}(y^n,x)=j \\}|$ is the number of domain elements with frequency ${j}$ in $y^{n}$. We call $n$ the length of profile $\\phi$ and use $\\Phi^n$ denote the set of all profiles of length $n$.\n\\footnote{The profile does not contain $\\phi(0)$, the number of unseen domain elements.}\n\\end{defn}\n\nFor any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the probability of a profile $\\phi \\in \\Phi^n$ is defined as:\n\\begin{equation}\\label{eqpml1}\n\\bbP(\\textbf{p},\\phi)\\eqdef\\sum_{\\{y^n \\in \\mathcal{D}^n~|~ \\Phi (y^n)=\\phi \\}} \\mathbb{P}(\\textbf{p},y^n)\\\\\n\\end{equation}\n\nThe distribution that maximizes the probability of a profile $\\phi$ is the profile maximum likelihood distribution and we formally define it next.\n\\begin{defn}[Profile maximum likelihood distribution] For any profile $\\phi \\in \\Phi^{n}$, a \\emph{Profile Maximum Likelihood} (PML) distribution $\\textbf{p}_{pml,\\phi} \\in \\Delta^{\\mathcal{D}}$ is:\n $\\textbf{p}_{pml,\\phi} \\in \\argmax_{\\textbf{p} \\in \\Delta^{\\mathcal{D}}} \\bbP(\\textbf{p},\\phi)$ and $\\bbP(\\textbf{p}_{pml,\\phi},\\phi)$ is the maximum PML objective value. Further, a distribution $\\textbf{p}^{\\beta}_{pml,\\phi} \\in \\Delta^{\\mathcal{D}}$ is a $\\beta$-\\emph{approximate PML} distribution if $\\bbP(\\textbf{p}^{\\beta}_{pml,\\phi},\\phi)\\geq \\beta \\cdot \\bbP(\\textbf{p}_{pml,\\phi},\\phi)$.\n\\end{defn}\n\nWe next provide formal definitions for separable symmetric property and an estimator.\n\\begin{defn}[Separable Symmetric Property] A symmetric property $\\textbf{f}:\\Delta^{\\mathcal{D}} \\rightarrow \\mathbb{R}$ is separable if for any $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, $f(\\textbf{p})\\eqdef\\sum_{x\\in \\mathcal{D}}\\textbf{g}(\\textbf{p}_{x})$, for some function $\\textbf{g}:\\mathbb{R} \\rightarrow \\mathbb{R}$. Further for any subset $S\\subset \\mathcal{D}$, we define $f_{S}(\\textbf{p})\\eqdef\\sum_{x\\in S}\\textbf{g}(\\textbf{p}_{x})$.\n\t\\end{defn}\n\n\\begin{defn}\n\tA property estimator is a function $\\hat{\\textbf{f}}:\\mathcal{D}^n \\rightarrow \\mathbb{R}$, that takes as input $n$ samples and returns the estimated property value. The sample complexity of $\\hat{\\textbf{f}}$ for estimating a symmetric property $\\textbf{f}(\\textbf{p})$ is the number of samples needed to estimate $\\textbf{f}$ up to accuracy $\\epsilon$ and with constant probability. The optimal sample complexity of a property $\\textbf{f}$ is the minimum number of samples of any estimator.\n\t\\end{defn} \n\\section{Main Results}\\label{sec:results}\nAs discussed in the introduction, one of our motivations was to provide a better analysis for the PML distribution based plug-in estimator. In this direction, we first show that the PML distribution is sample complexity optimal in estimating support in all parameter regimes. Estimating support is difficult in general and all previous works make the assumption that the minimum non-zero probability value of the distribution is at least $\\frac{1}{k}$. In our next result, we show that the PML distribution under this constraint is sample complexity optimal for estimating support.\n\\begin{thm}\\label{thm:supp}\n\tThe PML distribution \\footnote{\\label{suppfoot} Under the constraint that its minimum non-zero probability value is at least $\\frac{1}{k}$. This assumption is also necessary for the results in \\cite{ADOS16} to hold.} based plug-in estimator is sample complexity optimal in estimating support for all regimes of error parameter $\\epsilon$.\n\\end{thm}\nFor support, we show that an approximate PML distribution is sample complexity optimal as well.\n\\begin{thm}\\label{thm:approxsupp}\n\tFor any constant $\\alpha>0$, an $\\exps{-\\epsilon^2 n^{1-\\alpha}}$-approximate PML distribution \\footnoteref{suppfoot} based plug-in estimator is sample complexity optimal in estimating support for all regimes of error $\\epsilon$.\n\\end{thm}\nWe defer the proof of both these theorems to \\Cref{sec:support}.\n\nFor entropy and distance to uniformity, we study a variation of the PML distribution we call the pseudo PML distribution and present a general framework for symmetric property estimation based on this. We show that this pseudo PML based general approach gives an estimator that is sample complexity optimal for estimating entropy and distance to uniformity in broader parameter regimes. To motivate and understand this general framework we first define new generalizations of the profile, PML and approximate PML distributions.\n\\newcommand{\\phis(j)}{\\phi_{S}(j)}\n\\begin{defn}[$S$-pseudo Profile]\n\tFor any sequence $y^{n} \\in \\mathcal{D}^{n}$ and $S \\subseteq \\mathcal{D}$, its $S$-\\emph{pseudo} profile denoted $\\phi_{S}=\\Phi_{S}(y^n)$ is $\\phi\\eqdef(\\phis(j))_{j\\in[n]} $ where $\\phis(j)\\eqdef|\\{x\\in S ~|~\\textbf{f}(y^n,x)=j \\}|$ is the number of domain elements in $S$ with frequency ${j}$ in $y^{n}$. We call $n$ the length of $\\phi_{S}$ as it represents the length of the sequence $y^n$ from which this pseudo profile was constructed. Let $\\Phi_{S}^{n}$ denote the set of all $S$-pseudo profiles of length $n$.\n\\end{defn}\n\nFor any distribution $\\textbf{p} \\in \\Delta^{\\mathcal{D}}$, the probability of a $S$-pseudo profile $\\phi_{S} \\in \\Phi_{S}^{n}$ is defined as:\n\\begin{equation}\\label{eqpml1}\n\\bbP(\\textbf{p},\\phi_{S})\\eqdef\\sum_{\\{y^n \\in \\mathcal{D}^n~|~ \\Phi_{S}(y^n)=\\phi_{S} \\}} \\mathbb{P}(\\textbf{p},y^n)\\\\\n\\end{equation}\nWe next define the $S$-pseudo PML and $(\\beta,S)$-approximate pseudo PML distributions that are analogous to the PML and approximate PML distributions.\n\\begin{defn}[$S$-pseudo PML distribution]\n\tFor any $S$-pseudo profile $\\phi_{S} \\in \\Phi_{S}^{n}$, a distribution $\\textbf{p}_{\\phi_{S}} \\in \\Delta^{\\mathcal{D}}$ is a $S$-\\emph{pseudo PML} distribution if $\\textbf{p}_{\\phi_{S}} \\in \\argmax_{\\textbf{p} \\in \\Delta^{\\mathcal{D}}} \\bbP(\\textbf{p},\\phi_{S})$.\n\\end{defn}\n\n\\begin{defn}[$(\\beta,S)$-approximate pseudo PML distribution]\n\tFor any profile $\\phi_{S} \\in \\Phi_{S}^{n}$, a distribution $\\bp^{\\beta}_{\\phi_{S}} \\in \\Delta^{\\mathcal{D}}$ is a $(\\beta,S)$-\\emph{approximate pseudo PML} distribution if $\\bbP(\\bp^{\\beta}_{\\phi_{S}},\\phi_{S})\\geq \\beta \\cdot \\bbP(\\bp^{\\beta}_{\\phi_{S}},\\phi_{S})$.\n\\end{defn}\nFor notational convenience, we also define the following function.\n\\begin{defn}\n\tFor any subset $S \\subseteq \\mathcal{D}$, the function $\\mathrm{Freq}:\\Phi_{S}^{n} \\rightarrow 2^{\\mathbb{Z}_{+}}$ takes input a $S$-psuedo profile and returns the set with all distinct frequencies in $\\phi_{S}$.\n\\end{defn}\n\nUsing the definitions above, we next give an interesting generalization of Theorem 3 in \\cite{ADOS16}.\n\\begin{thm}\\label{thmbeta}\n\tFor a symmetric property $\\textbf{f}$ and $S \\subseteq \\mathcal{D}$, suppose there is an estimator $\\hat{\\textbf{f}}:\\Phi_{S}^{n} \\rightarrow \\mathbb{R}$, such that for any $\\textbf{p}$ and $\\phi_{S} \\sim \\textbf{p}$ the following holds, $$\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\hat{\\textbf{f}}(\\phi_{S})|\\geq \\epsilon} \\leq \\delta~,$$ then for any $\\mathrm{F} \\in 2^{\\mathbb{Z}_{+}}$, a $(\\beta,S)$-approximate pseudo PML distribution $\\bp^{\\beta}_{\\phi_{S}}$ satisfies: $$\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})| \\geq 2 \\epsilon} \\leq \\frac{\\delta n^{|\\mathrm{F}|}}{\\beta} \\Prob{\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}} +\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{F}}~.$$\n\\end{thm}\n\nNote that in the theorem above, the error probability with respect to a pseudo PML distribution based estimator has dependency on$\\frac{\\delta n^{|\\mathrm{F}|}}{\\beta}$ and $\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{F}}$. However Theorem 3 in \\cite{ADOS16} has error probability $\\frac{\\delta e^{\\sqrt{n}}}{\\beta}$. This is the bottleneck in showing that PML works for all parameter regimes and the place where pseudo PML wins over the vanilla PML based estimator, getting non-trivial results for entropy and distance to uniformity. We next state our general framework for estimating symmetric properties.\nWe use the idea of sample splitting which is now standard in the literature \\cite{WY16,JVHW15,JHW16,CL11,Nem03}.\n\\begin{algorithm}[H]\n\t\\caption{General Framework for Symmetric Property Estimation}\\label{algpml}\n\t\\begin{algorithmic}[1]\n\t\t\\Procedure{Property estimation}{$x^{2n},\\textbf{f}, \\mathrm{F}$}\n\t\t\\State Let $x^{2n}=(x_1^n,x_2^n)$, where $x_1^n$ and $x_2^n$ represent first and last $n$ samples of $x^{2n}$ respectively.\n\t\t\\State Define $S\\eqdef \\{y \\in \\mathcal{D}~|~f(x_1^n,y) \\in \\mathrm{F} \\}$.\n\t\t\\State Construct profile $\\phi_{S}$, where $\\phis(j)\\eqdef|\\{y\\in S ~|~\\textbf{f}(x_2^n,y)=j \\}|$.\n\t\t\\State Find a $(\\beta,S)$-approximate pseudo PML distribution $\\bp^{\\beta}_{\\phi_{S}}$ and empirical distribution $\\hat{\\textbf{p}}$ on $x_2^n$.\n\t\t\\State \\textbf{return} $\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})+\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}}) + \\text{correction bias with respect to }\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})$.\n\t\t\\EndProcedure\n\t\\end{algorithmic}\n\\end{algorithm}\nIn the above general framework, the choice of $\\mathrm{F}$ depends on the symmetric property of interest. Later, in the case of entropy and distance to uniformity, we will choose $\\mathrm{F}$ to be the region where the empirical estimate fails; it is also the region that is difficult to estimate. One of the important properties of the above general framework is that $\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})$ (recall $\\bp^{\\beta}_{\\phi_{S}}$ is a $(\\beta,S)$-approximate pseudo PML distribution and $\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})$ is the property value of distribution $\\bp^{\\beta}_{\\phi_{S}}$ on subset of domain elements $S \\subseteq \\mathcal{D}$) is close to $\\textbf{f}_{S}(\\textbf{p})$ with high probability. Below we state this result formally.\n\\begin{thm}\\label{thm:main}\n\tFor any symmetric property $\\textbf{f}$, let $\\mathrm{G} \\subseteq {\\mathcal{D}}$ and $\\mathrm{F},\\mathrm{\\bar{F}'} \\in 2^{\\mathbb{Z}_{+}}$. If for all $S' \\in 2^\\mathrm{G}$, there exists an estimator $\\hat{\\textbf{f}}:\\Phi_{S'}^{n} \\rightarrow \\mathbb{R}$, such that for any $\\textbf{p}$ and $\\phi_{S'} \\sim \\textbf{p}$ satisfies, \n\t\\begin{equation}\\label{eq:condio}\n\t\\Prob{|\\textbf{f}_{{S'}}(\\textbf{p})-\\hat{\\textbf{f}}(\\phi_{S'})|\\geq2 \\epsilon} \\leq \\delta \\text{ and }\\Prob{\\dfreq{\\phi_{S'}} \\not\\subseteq \\mathrm{F}'} \\leq \\gamma~.\n\t\\end{equation}\n\tThen for any sequence $x^{2n}=(x_1^n,x_2^n)$, $$\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>4\\epsilon}\\leq \\frac{\\delta n^{|F'|}}{\\beta}+\\gamma + \\Prob{S \\notin 2^\\mathrm{G}}~,$$ where $S$ is a random set $\\textbf{S}\\eqdef \\{y \\in \\mathcal{D}~|~f(x_1^n,y) \\in \\mathrm{F} \\}$ and $\\phi_{S}\\eqdef \\Phi_{S}(x_2^n)$.\n\\end{thm}\n\nUsing the theorem above, we already have a good estimate for $\\textbf{f}_{S}(\\textbf{p})$ for appropriately chosen frequency subsets $\\mathrm{F}, \\mathrm{\\bar{F}'}$ and $\\mathrm{G} \\subseteq \\mathcal{D}$. Further, we choose these subsets $\\mathrm{F}, \\mathrm{\\bar{F}'}$ and $\\mathrm{G}$ carefully so that the empirical estimate $\\textbf{f}_{\\bar{S}}(\\hat{p})$ plus the correction bias with respect to $\\textbf{f}_{\\bar{S}}$ is close to $\\textbf{f}_{\\bar{S}}(\\textbf{p})$. Combining these together, we get the following results for entropy and distance to uniformity. \n\\begin{thm}\\label{thm:entr}\n\tIf error parameter $\\epsilon>\\Omega\\left(\\frac{\\log N}{N^{1-\\alpha}}\\right)$ for any constant $\\alpha>0$,\n\tthen for estimating entropy, the estimator \\ref{algpml} for $\\beta=n^{-\\log n}$ is sample complexity optimal.\n\\end{thm}\nFor entropy, we already know from \\cite{WY16} that the empirical distribution is sample complexity optimal if $\\epsilon 0$. Therefore the interesting regime for entropy estimation is when $\\epsilon>\\Omega\\left(\\frac{\\log N}{N}\\right)$ and our estimator works for almost all such $\\epsilon$.\n\\begin{thm}\\label{thm:dtu}\n\tLet $\\alpha>0$ and error parameter $\\epsilon>\\Omega\\left(\\frac{1}{N^{1-8\\alpha}}\\right)$, then for estimating distance from uniformity, the estimator \\ref{algpml} for $\\beta=n^{-\\sqrt{\\frac{n \\log n}{N}}}$ is sample complexity optimal.\n\\end{thm}\nNote that the estimator in \\cite{JHW17} also requires that the error parameter $\\epsilon \\geq \\frac{1}{N^C}$, where $C>0$ is some constant.\n\\section{Analysis of General Framework for Symmetric Property Estimation}\\label{app:universal}\nHere we provide proofs of the main results for our general framework (Theorem`\\ref{thmbeta} and \\ref{thm:main}). These results weakly depend on the property and generalize results in \\cite{ADOS16}. The PML based estimator in \\cite{ADOS16} is sample competitive only for a restricted error parameter regime and this stems from the large number of possible profiles of length $n$. Our next lemma will be useful to address this issue and later we show how to use this result to prove Theorems \\ref{thmbeta} and \\ref{thm:main}.\n\n\\begin{lemma}\\label{lem:bcard}\n\t For any subset $S \\subseteq \\mathcal{D}$ and $\\mathrm{F} \\in 2^{\\mathbb{Z}_{+}}$, if set $B$ is defined as $B\\eqdef \\{\\phi_{S} \\in \\Phi_{S}^{n}~|~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F} \\}$, then the cardinality of set $B$ is upper bounded by $(n+1)^{|\\mathrm{F}|}$.\n\t\\end{lemma}\n\\begin{proof}[Proof of \\Cref{thmbeta}]\n\tUsing the law of total probability we have,\n\t\\begin{align*}\n\t\\begin{split}\n\t\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon} &=\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon~,~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}}\\\\\n\t&\\quad + \\Prob{ |\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon~,~\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{F}},\\\\\n\t& \\leq \\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon~,~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}} + \\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{F}}~.\n\t\\end{split}\n\t\\end{align*}\n\t\n\tConsider any $\\phi_{S} \\sim \\textbf{p}$. If $\\probbp{\\phi_{S}}>\\delta\/\\beta$, then we know that $\\probbpml{\\phi_{S}}>\\delta$. For $\\beta \\leq 1$, we have $\\probbp{\\phi_{S}}>\\delta$ that implies $|\\textbf{f}_{S}(\\textbf{p})-\\estiS{\\phi_{S}}| \\leq \\epsilon$. Further $\\probbpml{\\phi_{S}}>\\delta$ implies $|\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})-\\estiS{\\phi_{S}}| \\leq \\epsilon$. Using triangle inequality we get,\n\t$|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})| \\leq |\\textbf{f}_{S}(\\textbf{p})-\\estiS{\\phi_{S}}|+|\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})-\\estiS{\\phi_{S}}| \\leq 2\\epsilon$.\n\tNote we wish to upper bound the probability of set: $\\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}}\\eqdef \\{\\phi_{S} \\in \\Phi_{S}^{n}~|~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F} \\text{ and }|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2 \\epsilon\\}$. From the previous discussion, we get $\\probbp{\\phi_{S}} \\leq \\delta\/\\beta$ for all $\\phi_{S} \\in \\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}}$. Therefore,\n\t\\begin{align*}\n\t\\Prob{|\\textbf{f}_{{S}}(\\textbf{p})-\\textbf{f}_{{S}}(\\bp^{\\beta}_{\\phi_{S}})|\\geq 2\\epsilon~,~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}}=\\sum_{\\phi_{S} \\in \\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}} }\\probbp{\\phi_{S}} \\leq \\frac{\\delta}{\\beta}|\\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}}| \\leq \\frac{\\delta}{\\beta} (n+1)^{|\\mathrm{F}|}~.\n\t\\end{align*}\n\tIn the final inequality, we use $\\textbf{B}_{\\mathrm{F},S,\\hat{\\bff}} \\subseteq \\{\\phi_{S} \\in \\Phi_{S}^{n}~|~\\dfreq{\\phi_{S}} \\subseteq \\mathrm{F}\\}$ and invoke \\Cref{lem:bcard}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof for \\Cref{thm:main}]\nUsing Bayes rule we have:\n\t\\begin{equation}\n\t\\begin{split}\n\t\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon}&=\\sum_{S'\\subseteq \\mathcal{D}} \\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon~|~S=S'}\\Prob{S=S'}\\\\\n\t&\\leq \\sum_{S' \\in 2^\\mathrm{G}} \\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon~|~S=S'}\\Prob{S=S'}+ \\Prob{S \\notin 2^\\mathrm{G}}~.\n\t\\end{split}\n\t\\end{equation}\n\tIn the second inequality, we use $\\sum_{S' \\notin 2^\\mathrm{G}} \\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon ~,~ S=S'} \\leq \\Prob{S \\notin 2^\\mathrm{G}}$. Consider the first term on the right side of the above expression and note that it is upper bounded by, $\\sum_{S' \\in 2^\\mathrm{G}} \\Prob{|\\textbf{f}_{S'}(\\textbf{p})-\\textbf{f}_{S'}(\\bp_{\\phi_{S'}}^{\\beta})|>2\\epsilon}\\Prob{S=S'}\n\\leq \\sum_{S' \\in 2^\\mathrm{G}} \\left[\\frac{\\delta n^{|F'|}}{\\beta}+\\Prob{\\dfreq{\\phi_{S'}} \\not\\subseteq \\mathrm{\\bar{F}'}} \\right] \\Prob{S=S'}\n\\leq \\frac{\\delta n^{|F'|}}{\\beta}+\\gamma$.\n\tIn the first upper bound, we removed randomness associated with the random set $S$ and used $\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon~|~S=S'}= \\Prob{|\\textbf{f}_{S'}(\\textbf{p})-\\textbf{f}_{S'}(\\bp_{\\phi_{S'}}^{\\beta})|>2\\epsilon}$.\n\tIn the first inequality above, we invoke \\Cref{thmbeta} using conditions from \\Cref{eq:condio}. In the second inequality, we use $\\sum_{S' \\in 2^\\mathrm{G}} \\Prob{S=S'} \\leq 1$ and $\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}, S=S'} \\leq \\gamma$. The theorem follows by combining all the analysis together.\n\\end{proof}\n \n \n \\section{Applications of the General Framework}\\label{sec:appl}\nHere we provide applications of our general framework (defined in \\Cref{sec:results}) using results from the previous section. We apply our general framework to estimate entropy and distance to uniformity. In \\Cref{sec:entr} and \\Cref{sec:dtu} we analyze the performance of our estimator for entropy and distance to uniformity estimation respectively.\n\\subsection{Entropy estimation}\\label{sec:entr}\nIn order to prove our main result for entropy (\\Cref{thm:entr}), we first need the existence of an estimator for entropy with some desired properties. The existence of such an estimator will be crucial to bound the failure probability of our estimator. A result analogous to this is already known in \\cite{ADOS16} (Lemma 2) and the proof of our result follows from a careful observation of \\cite{ADOS16, WY16}. We state this result here but defer the proof to appendix.\n\\begin{lemma}\\label{thm:entrochange}\n\tLet $\\alpha>0$, $\\epsilon>\\Omega\\left(\\frac{\\log N}{N^{1-\\alpha}}\\right)$ and $S \\subseteq \\mathcal{D}$, then for entropy on subset $S$ ($\\sum_{y\\in S}\\textbf{p}_{y}\\log \\frac{1}{\\textbf{p}_{y}}$) there exists an $S$-pseudo profile based estimator that use the optimal number of samples, has bias less than $\\epsilon$ and if we change any sample, changes by at most $c\\cdot \\frac{n^{\\alpha}}{n}$, where $c$ is a constant.\n\\end{lemma}\nCombining the above lemma with \\Cref{thm:main}, we next prove that our estimator defined in \\Cref{algpml} is sample complexity optimal for estimating entropy in a broader regime of error $\\epsilon$.\n\\begin{proof}[Proof for \\Cref{thm:entr}]\n\tLet $\\textbf{f}(\\textbf{p})$ represent the entropy of distribution $\\textbf{p}$ and $\\hat{\\textbf{f}}$ be the estimator in \\Cref{thm:entrochange}.\n\tDefine $\\mathrm{F} \\eqdef [0, c_{1} \\log n]$ for constant $c_{1}\\geq 40$. Given the sequence $x^{2n}$, the random set $S$ is defined as $S\\eqdef \\{y \\in \\mathcal{D}~|~f(x_1^n,y) \\leq c_{1} \\log n \\}$.\n\tLet $\\mathrm{\\bar{F}'}\\eqdef[0,8c_{1} \\log n]$, then by derivation in Lemma 6~\\cite{ADOS16} (or by simple application of Chernoff \\footnote{Note probability of many events in this proof can be easily bounded by application of Chernoff. These bounds on probabilities are also shown in \\cite{ADOS16, WY16} and we use these inequalities by omitting details.}) we have,\n\t$$ \\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}} =\\Prob{\\exists y\\in \\mathcal{D} \\text{ such that }\\textbf{f}(x_1^n,y)\\leq c_{1} \\log n \\text{ and }\\textbf{f}(x_2^n,y)> 8c_{1} \\log n}\\leq \\frac{1}{n^{5}}~.$$\n\tFurther let $\\mathrm{G}\\eqdef \\{x \\in \\mathcal{D} ~|~\\textbf{p}_{x}\\leq \\frac{2c_{1} \\log N}{n}\\}$, then by Equation 48 in \\cite{WY16} we have, $\\Prob{S \\notin 2^\\mathrm{G}} \\leq \\frac{1}{n^4}$.\n\tFurther for all $S' \\in 2^{\\mathrm{G}}$ we have,\n\t$$\\Prob{\\dfreq{\\phi_{S'}} \\not\\subseteq \\mathrm{\\bar{F}'}}=\\Prob{\\exists y\\in S' \\text{ such that }\\textbf{f}(x_2^n,y)> 8c_{1} \\log n}\\leq \\gamma \\text{ for } \\gamma=\\frac{1}{n^{5}}~.$$\n\tNote for all $x \\in S'$, $\\textbf{p}_{x} \\leq \\frac{2c_{1} \\log N}{n}$ and the above inequality also follows from Chernoff. All that remains now is to upper bound $\\delta$. Using the estimator constructed in \\Cref{thm:entrochange} and further combined with McDiarmid's inequality, we have,\n\t$$\\Prob{|\\textbf{f}_{{S'}}(\\textbf{p})-\\hat{\\textbf{f}}(\\phi_{S'})|\\geq 2 \\epsilon} \\leq 2\\expo{\\frac{-2\\epsilon^2}{n(c\\frac{n^\\alpha}{n})^2}} \\leq \\delta \\text{ for }\\delta= \\expo{-2\\epsilon^2n^{1-2\\alpha}}~.$$\nSubstituting all these parameters together in \\Cref{thm:main} we have,\n\t\\begin{equation}\n\t\\begin{split}\n\t\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon}&\\leq \\frac{\\delta n^{|F'|}}{\\beta}+\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}} + \\Prob{S \\notin 2^\\mathrm{G}}\\\\\n\t&\\leq \\expo{-2\\epsilon^2n^{1-2\\alpha}} n^{9 c_{1} \\log n}+\\frac{1}{n^4} \\leq \\frac{2}{n^4}~.\n\t\\end{split}\n\t\\end{equation}\n\tIn the first inequality, we use \\Cref{thm:main}. In the second inequality, we substituted the values for $\\delta, \\gamma, \\beta$ and $\\Prob{S \\notin 2^\\mathrm{G}}$. In the final inequality we used $n=\\Theta(\\frac{N}{\\log N}\\frac{1}{\\epsilon})$ and $\\epsilon>\\Omega\\left(\\frac{\\log^3 N}{N^{1-4\\alpha}}\\right)$.\n\t\n\t\tOur final goal is to estimate $\\textbf{f}(\\textbf{p})$, and to complete the proof we need to argue that $\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})$ + the correction bias with respect to $\\textbf{f}_{\\bar{S}}$ is close to $\\textbf{f}_{\\bar{S}}(\\textbf{p})$, where recall $\\hat{\\textbf{p}}$ is the empirical distribution on sequence $x_2^n$. The proof for this follows immediately from \\cite{WY16} (Case 2 in the proof of Proposition 4). \\cite{WY16} bound the bias and variance of the empirical estimator with a correction bias and applying Markov inequality on their result we get $\\Prob{|\\textbf{f}_{\\bar{S}}(\\textbf{p})-(\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})+\\frac{|\\bar{S}|}{n})|>2\\epsilon} \\leq \\frac{1}{3}$, where $\\frac{|\\bar{S}|}{n}$ is the correction bias in \\cite{WY16}. Using triangle inequality, our estimator fails if either $|\\textbf{f}_{\\bar{S}}(\\textbf{p})-(\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})+\\frac{|\\bar{S}|}{n})|>2\\epsilon$ or $|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon$. Further by union bound the failure probability is at most $\\frac{1}{3}+\\frac{2}{n^4}$, which is a constant.\n\\end{proof}\n\\subsection{Distance to Uniformity estimation}\\label{sec:dtu}\nHere we prove our main result for distance to uniformity estimation (\\Cref{thm:dtu}). First, we show existence of an estimator for distance to uniformity with certain desired properties. Similar to entropy, a result analogous to this is shown in \\cite{ADOS16} (Lemma 2) and the proof of our result follows from the careful observation of \\cite{ADOS16, JHW17}. We state this result here but defer the proof to \\Cref{sec:omitted}.\n\\begin{lemma}\\label{thm:dtuchange}\n\tLet $\\alpha>0$ and $S \\subseteq \\mathcal{D}$, then for distance to uniformity on $S$ ($\\sum_{y \\in S}|\\textbf{p}_{y}-\\frac{1}{N}|$) there exists an $S$-pseudo profile based estimator that use the optimal number of samples, has bias at most $ \\epsilon$ and if we change any sample, changes by at most $c\\cdot \\frac{n^{\\alpha}}{n}$, where $c$ is a constant.\n\\end{lemma}\nCombining the above lemma with \\Cref{thm:main} we provide the proof for \\Cref{thm:dtu}.\n\\begin{proof}[Proof for \\Cref{thm:dtu}]\n\tLet $\\textbf{f}(\\textbf{p})$ represent the distance to uniformity for distribution $\\textbf{p}$ and $\\hat{\\textbf{f}}$ be the estimator in \\Cref{thm:dtuchange}.\n\tDefine $\\mathrm{F} = [\\frac{n}{N}-\\sqrt{\\frac{c_{1}n \\log n}{N}}, \\frac{n}{N}+\\sqrt{\\frac{c_{1}n \\log n}{N}}]$ for some constant $c_1 \\geq 40$. Given the sequence $x^{2n}$, the random set $S$ is defined as $S\\eqdef \\{y \\in \\mathcal{D}~|~f(x_1^n,y) \\in \\mathrm{F} \\}$. Let $\\mathrm{\\bar{F}'}=[\\frac{n}{N}-\\sqrt{\\frac{8c_{1}n \\log n}{N}}, \\frac{n}{N}+\\sqrt{\\frac{8c_{1} n\\log n}{N}}]$, then by derivation in Lemma 7 of \\cite{ADOS16} (also shown in \\cite{JHW17} \\footnote{Similar to entropy, for many events their probabilities can be bounded by simple application of Chernoff and have already been shown in \\cite{ADOS16, JHW17}. We omit details for these inequalities.}) we have,\n\t$$ \\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}} =\\Prob{\\exists y\\in \\mathcal{D} \\text{ such that }\\textbf{f}(x_1^n,y)\\in \\mathrm{F} \\text{ and }\\textbf{f}(x_2^n,y) \\notin \\mathrm{\\bar{F}'}}\\leq \\frac{1}{n^{4}}~.$$\n\t Further let $\\mathrm{G}\\eqdef \\{x \\in \\mathcal{D} ~|~\\textbf{p}_{x}\\in [\\frac{1}{N}-\\sqrt{\\frac{2c_{1} \\log n}{nN}}, \\frac{1}{N}+\\sqrt{\\frac{2c_{1} \\log n}{nN}}]\\}$, then using Lemma 2 in \\cite{JHW17} we get,\n\t$$\\Prob{S \\notin 2^\\mathrm{G}}=\\Prob{\\exists y \\in \\mathcal{D} \\text{ such that }\\textbf{f}(x_1^n,y)\\in \\mathrm{F} \\text{ and }\\textbf{p}_{x} \\notin G} \\leq \\frac{\\log n}{n^{1-\\epsilon}}~.$$\n\tFurther for all $S' \\in 2^{\\mathrm{G}}$ we have,\n\t$$\\Prob{\\dfreq{\\phi_{S'}} \\not\\subseteq \\mathrm{\\bar{F}'}}=\\Prob{\\exists y\\in S' \\text{ such that }\\textbf{f}(x_2^n,y)> 8c_{1} \\log n}\\leq \\gamma \\text{ for } \\gamma=\\frac{1}{n}~.$$\n\tNote for all $x \\in S'$, $\\textbf{p}_{x} \\in G$ and the above result follows from \\cite{JHW17} (Lemma 1). All that remains now is to upper bound $\\delta$. Using the estimator constructed in \\Cref{thm:dtuchange} and further combined with McDiarmid's inequality, we have,\n\t$$\\Prob{|\\textbf{f}_{{S'}}(\\textbf{p})-\\hat{\\textbf{f}}(\\phi_{S'})|\\geq 2 \\epsilon} \\leq 2\\expo{\\frac{-2\\epsilon^2}{n(c\\frac{n^\\alpha}{n})^2}} \\leq \\delta \\text{ for }\\delta= \\expo{-2\\epsilon^2n^{1-2\\alpha}}~.$$\n\tSubstituting all these parameters in \\Cref{thm:main} we get,\n\t\\begin{equation}\\label{eq:entropml}\n\t\\begin{split}\n\t\\Prob{|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon}&\\leq \\frac{\\delta n^{|F'|}}{\\beta}+\\Prob{\\dfreq{\\phi_{S}} \\not\\subseteq \\mathrm{\\bar{F}'}} + \\Prob{S \\notin 2^\\mathrm{G}}\\\\\n\t&\\leq \\expo{-2\\epsilon^2n^{1-2\\alpha}} n^{2\\sqrt{\\frac{8c_{1}n \\log n}{N}}}+\\frac{\\log n}{n^{1-\\epsilon}}+\\frac{1}{n} \\leq o(1)~.\n\t\\end{split}\n\t\\end{equation}\n\tIn the first inequality, we use \\Cref{thm:main}. In the second inequality, we substituted values for $\\delta, \\gamma, \\beta$ and $\\Prob{S \\notin 2^\\mathrm{G}}$. In the final inequality we used $n=\\Theta(\\frac{N}{\\log N}\\frac{1}{\\epsilon^2})$ and $\\epsilon>\\Omega\\left(\\frac{1}{N^{1-8\\alpha}}\\right)$.\n\t\n\tOur final goal is to estimate $\\textbf{f}(\\textbf{p})$, and to complete the proof we argue that $\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})$ + correction bias with respect to $\\textbf{f}_{\\bar{S}}$ is close to $\\textbf{f}_{\\bar{S}}(\\textbf{p})$, where recall $\\hat{\\textbf{p}}$ is the empirical distribution on sequence $x_2^n$. The proof for this case follows immediately from \\cite{JHW17} (proof of Theorem 2). \\cite{JHW17} define three kinds of events $\\mathcal{E}_{1},\\mathcal{E}_{2}$ and $\\mathcal{E}_{3}$, the proof for our empirical case follows from the analysis of bias and variance of events $\\mathcal{E}_{1}$ and $\\mathcal{E}_{2}$. Further combining results in \\cite{JHW17} with Markov inequality we get $\\Prob{|\\textbf{f}_{\\bar{S}}(\\textbf{p})-\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})|>2\\epsilon} \\leq \\frac{1}{3}$, and the correction bias here is zero. Using triangle inequality, our estimator fails if either $|\\textbf{f}_{\\bar{S}}(\\textbf{p})-(\\textbf{f}_{\\bar{S}}(\\hat{\\textbf{p}})+\\frac{|\\bar{S}|}{n})|>2\\epsilon$ or $|\\textbf{f}_{S}(\\textbf{p})-\\textbf{f}_{S}(\\bp^{\\beta}_{\\phi_{S}})|>2\\epsilon$. Further by union bound the failure probability is upper bounded by $\\frac{1}{3}+o(1)$, which is a constant.\n\\end{proof}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}