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+{"text":"\\section{Introduction}\n\nA holomorphic motion in dimension one is a family of injections $f_{\\lambda}:A\\to \\hat{\\mathbb C}$ over a complex manifold $\\Lambda\\ni\\lambda$.  Holomorphic motions first appeared in \\cite{MSS,L} where they were used to show that a generic rational map $f:\\hat{\\mathbb C}\\to\\hat{\\mathbb C}$ is structurally stable.  This notion has since found numerous applications in holomorphic dynamics and  Teichm\\\"{u}ller Theory. \nIts usefulness comes from the fact that analyticity alone forces strong extendibility and regularity properties that are referred to as the $\\lambda$-lemma. Let $\\Delta$ be the unit disk in $\\mathbb C$. \n\n\\begin{theorem}$ $\n\\begin{itemize}\n\\item {\\bf Extension $\\lambda$-lemma} \\cite{L}, \\cite{MSS} Any holomorphic motion $f:\\Delta\\times A \\to \\hat\\C$ extends to a holomorphic motion  $\\Delta\\times \\bar A \\ra \\hat \\C$. \n\n\\item {\\bf QC $\\lambda$-lemma} \\cite{MSS} The map $f(\\lambda,a)$ is uniformly quasisymmetric in $a$.\n\\end{itemize}\n\\end{theorem}\n\nNote that when $A$ has interior, $f(\\lambda, a)$ is quasiconformal on the interior. For many applications it is important to know that a holomorphic motion can be extended to a holomorphic motion of the entire sphere.\nBers \\& Royden \\cite{BR} and Sullivan \\& Thurston\n\\cite{ST} proved that there exists a universal $\\delta>0$ such\nthat under the circumstances of the Extension $\\lambda$-lemma, the\nrestriction of $f$ to the parameter disk $\\Delta_{\\delta}$ of\nradius $\\delta$ can be extended to a holomorphic motion\n$\\Delta_{\\delta} \\times \\hat {{\\mathbb C}} \\mapsto \\hat{\\mathbb\nC}$. S\\l{}odkowski \n\\cite{Slodkowski} proved the strongest version asserting that $\\delta$ is\nactually equal to $1$:\n\n\\newtheorem*{Slod lemma}{$\\lambda$-lemma [S\\l{o}dkowski]}\n\\begin{Slod lemma}\nLet  $A\\subset \\hat{\\mathbb C}$. Any holomorphic motion\n$f:\\Delta\\times A \\to \\hat{\\mathbb C}$ extends to a holomorphic\nmotion $\\Delta\\times \\hat{\\mathbb C} \\mapsto \\hat{\\mathbb C}$.\n\\end{Slod lemma}\n\nS\\l{}odkowski's proof builds on the work by Forstneri\\v{c} \\cite{Forstneric} and \\v{S}nirel'man  \\cite{Snirelman}.\nAstala and Martin \\cite{AM} gave an exposition of S\\l{}odkowski's proof from the point of view of $1$-dimensional complex analysis. Chirka \\cite{ChirkaOka} gave an independent proof using solution to $\\bar{\\partial}$-equation. (See \\cite{Teich} for a detailed exposition of Chirka's proof.) The purpose of this paper is to give a more geometric approach to the proof of the $\\lambda$-lemma. \nWe take S\\l{}odkowski's approach and replace the major technical part in his proof (closedness, see \\cite[Theorem 4.1]{AM}) by a geometric pseudoconvexity argument.\n\nThe strongest $\\lambda$-lemma fails when the dimension of the base\nmanifold is greater than $1$, even if the base is topologically contractible. This follows from the results of Earl-Kra \\cite{EK} and Hubbard \\cite{Hu}.\n\nWe give the necessary background on holomorphic motions, pseudoconvexity and Hilbert transform in Section \\ref{sec:background}. In Section \\ref{sec:axiom_of_choice}, we show that the $\\lambda$-lemma when $A$ is finite implies the $\\lambda$-lemma for arbitrary $A$. We set up the notations and terminology in Section \\ref{sec:terminology}. We state the filling theorem for the torus, and explain how it implies the finite $\\lambda$-lemma in Section \\ref{sec:filling_theorem}. In Section \\ref{sec:trapping_disks} we prove H\\\"{o}lder estimates for disks trapped inside pseudoconvex domains and construct such trapping pseudoconvex domains for ``graphical tori''. We use these estimates to prove the filling theorem in Section \\ref{sec:proof}.\n\\subsection{Acknowledgments} We would like to thank Misha Lyubich, Yakov Eliashberg and the referee for fruitful discussions and useful suggestions.\n\n\n\\section{Background}\\label{sec:background}\n\n\\subsection{Holomorphic motion}\n\n\nLet $\\Delta$ be a unit disk. Let $A\\subset \\hat{\\mathbb C}$. A {\\it holomorphic motion} of  $A$ is a map\n$f$: $\\Delta\\times A \\to \\hat{\\mathbb C}$ such that\n\\begin{enumerate}\n\\item for fixed $a\\in A$, the map $\\lambda\\mapsto f(\\lambda,a)$ is holomorphic in $\\Delta$\n\\item for fixed $\\lambda\\in \\Delta$, the map $a\\mapsto f(\\lambda,a)=:f_{\\lambda}(a)$ is an injection and\n\\item the map $f_0$ is the identity on $A$.\n\\end{enumerate}\n\n\\subsection{Pseudoconvexity}\\label{sec:pseudo}\n\nBelow we give definitions that are sufficient for our purposes.\n\nA $C^2$ smooth function is {\\it (strictly) plurisubharmonic} (written (strictly) psh) if its restriction to every complex line is strictly subharmonic. In coordinates \n$z=(z_1,\\dots,z_n)$, $u(z)$ is strictly psh if the matrix $\\left(\\frac{\\partial^2 u}{\\partial z_j\\partial \\bar{z}_k}\\right)$ is positive definite.\n\nA smoothly bounded domain $\\Omega\\subset \\mathbb C^2$ is {\\it strictly pseudoconvex} if there is a smooth, strictly psh\nfunction $\\rho$ in a neighborhood of $\\bar{\\Omega}$ such that $\\{\\Omega=\\rho(z)<0\\}.$\n\n\\begin{lemma}\\label{lem:disk_trapping}Let $\\Omega_s\\subset \\mathbb C^2$ be a\nfamily of pseudoconvex domains with defining functions $\\rho_s$,\n$s\\in[0,1]$. We assume that the family $\\rho_s$ is continuous in\n$s$. Let $\\phi_s:\\Delta\\mapsto \\mathbb C^2$ be a continuous family\nof holomorphic non-constant functions that extend continuously to\n$\\bar{\\Delta}$. Set $D_s:=\\phi_s(\\Delta)$. Suppose $\\partial\nD_s\\subset\n\\partial \\Omega_s$, $s\\in [0,1]$. And suppose $D_s\\subset\n\\Omega_s,$ $s\\in [0,1)$. Then $D_1\\subset \\Omega_1$.\n\\end{lemma}\n\n\\begin{proof} Consider the restriction of the functions $\\rho_s$ to $D_s$. The functions $\\rho_s\\circ \\phi_s:\\Delta\\mapsto \\mathbb R$ are subharmonic functions, $\\rho_1\\circ \\phi_1$ is the limit of $\\rho_s\\circ \\phi_s$. By the hypothesis of the lemma, $\\rho_s\\circ \\phi_s\\leq 0$ on $\\Delta$. Therefore,\n$\\rho_1\\circ \\phi_1\\leq 0$. If the maximum value $0$ is attained\nin the interior point, $\\rho_1\\circ \\phi_1\\equiv 0$. It implies\nthat $D_1\\subset \\partial \\Omega_1$, which is impossible.\nTherefore, $\\rho_1\\circ\\phi_1<0$ on $\\Delta$, and $D_1\\subset\n\\Omega_1$.\n\\end{proof}\n\nLet $M\\subset \\mathbb C^2$ be a real two-dimensional\nmanifold. We say that $p\\in M$ is a {\\it totally real} point if\n$T_pM\\cap iT_pM=\\{0\\}$. $M$ is a totally real manifold if all its\npoints are totally real. If the manifold $M$ is totally real, it is in fact homeomorphic to the torus (see \\cite{Bishop} and \\cite{GW}).\nAssume $M\\subset \\partial \\Omega$, then one can define a\ncharacteristic field of directions on $M$.\n\nLet $p\\in M$. Let $H_p\\partial \\Omega:=T_p\\Omega\\cap iT_p\\Omega$ be the holomorphic tangent space.\n$\\langle\\xi_p\\rangle:=H_p\\partial \\Omega\\cap T_pM$ is called the {\\it characteristic\ndirection}. We denote by $\\chi(M, \\Omega)$ the characteristic field of directions\n(see \\cite[Section 16.1]{CE}).\n\n\\subsection{Hilbert transform}\n\nA function $u:\\mathbb S^1\\to \\mathbb C$ is {\\it H\\\"{o}lder continuous with exponent} $\\alpha$\nif there is a constant $A$ such that for all $x,y\\in \\mathbb S^1$:\n$$|u(x)-u(y)|<A |x-y|^{\\alpha}.$$\n\nWe will consider the space $C^{1,\\alpha}(\\mathbb S^1)$ of differentiable functions $u$ with $\\alpha$- H\\\"{o}lder continuous derivative. The norm on the space $C^{1, \\alpha}(\\mathbb S^1)$ is defined by the formula:\n\n$$||u||_{1, \\alpha}:=\\sup_{x\\in \\mathbb S^1}|u(x)|+\\sup_{x\\in \\mathbb S^1}|u'(x)|+\\sup_{x\\neq y\\in \\mathbb S^1}\\frac{|u'(x)-u'(y)|}{|x-y|^{\\alpha}}.$$\n\n\nThere exists a unique harmonic extension $u_h$ of the function $u$ to $\\Delta$. Let denote by $u_h^*$ the harmonic conjugate of $u_h$, normalized by the condition $u_h(0)=0$. The function $u_h^*$ extends to $\\mathbb S^1=\\partial \\Delta$ as a H\\\"{o}lder continuous function with exponent\n$\\alpha$.\n\nFor a function $u\\in C^{1, \\alpha}(\\mathbb S^1)$ we define its {\\it Hilbert transform} $Hu$ to be the boundary value of the harmonic conjugate function $u_h^*$. By definition, the function $u+iHu$ \nextends as a holomorphic function to the unit disk.\n\n\\begin{theorem} The Hilbert transform $H$ is a bounded linear operator on $C^{1,\\alpha}(\\mathbb S^1)$ and $C^{\\alpha}(\\mathbb S^1)$.\n\\end{theorem}\nThis Theorem makes it convenient for us to work with the spaces $C^{1,\\alpha}(\\mathbb S^1)$ and $C^{\\alpha}(\\mathbb S^1)$.\n\n\\section{Finite $\\lambda$ -lemma}\\label{sec:axiom_of_choice}\nThe first step in the proof of the $\\lambda$-lemma is to reduce it\nto the $\\lambda$-lemma for finitely many points, \\cite{MSS}.\n\n\\begin{theorem}{\\bf The Finite $\\lambda$-lemma} Assume $a_1, \\dots, a_{n+1}\\in \\hat{\\mathbb C}$, $a_i\\neq a_j$ for $i\\neq j$. Let\n$f:\\Delta\\times\\{a_1, \\dots, a_n\\}\\to \\hat{\\mathbb C}$ be a holomorphic motion. Then there exists a\nholomorphic motion $\\tilde{f}: \\Delta\\times\\{a_1,\\dots, a_{n+1}\\}\\to \\mathbb C$, so that\n$\\tilde{f}$ is an extension of $f$.\n\\end{theorem}\n\n\\begin{proof}[Reduction of the $\\lambda$-lemma to the finite $\\lambda$-lemma (assuming the Extension-$\\lambda$ lemma):]\nWe normalize the holomorphic motion $f$ so that three points $a_1, a_2, a_3$ stay fixed. We can assume $a_1=0$, $a_2=1$, $a_3=\\infty$.\n\nLet $\\{a_n\\}$ be a sequence of points that are dense in $\\bar{A}$.\nLet $\\{z_n\\}$ be a sequence of points that are dense in $\\mathbb C\\backslash \\bar{A}$. Let $f_n$ be a holomorphic motion of $a_1,\\dots, a_n,z_1,\\dots, z_n$, such that\n$$f_n(\\lambda,a_i)=f(\\lambda,a_i).$$\nThe existence of such holomorphic motion follows from the Finite\n$\\lambda$-lemma.\n\nFor any fixed $z_n$, for $k\\geq n$ and $n\\geq 3,$ maps $f_k$ are defined at the\npoint $z_n$, and functions $f_k(*, z_n):\\Delta\\to \\mathbb\nC\\backslash\\{0,1\\}$ form a normal family. So we can choose a\nconvergent subsequence $f_{k}(*,z_n)$. Using the diagonal method, we get\na holomorphic motion $\\tilde{f}$, that is well defined for all\n$a_n$ and $z_n$ and coincides with $f$ on $a_i$ for all $i$. By the Extension\n$\\lambda$-lemma, we extend it uniquely to the holomorphic motion\nof $\\hat{\\mathbb{C}}$. By construction,  it coincides with the\nholomorphic motion $f$ on the set $A$.\n\\end{proof}\n\n\\section{Notations and Terminology}\\label{sec:terminology}\n\nWe consider $\\mathbb C^2$ with coordinates $(\\lambda,w)$. The horizontal direction is parametrized by $\\lambda$, the vertical by $w$.\nThroughout the paper we consider disks of the form\n$$w=g(\\lambda)$$\nthat will depend on two different parameters.\nWe will use the following notations\n$$g:\\Delta\\times\\mathbb S^1\\times [0,t_0]\\to \\mathbb C^2$$\n$$g^t_{\\xi}(\\lambda):=g^t(\\lambda,\\xi):=g_{\\xi}(\\lambda,t):=g(\\lambda,\\xi,t).$$\n\n\n\n\n\\subsection{Graphical Torus}\n\nLet $\\pi:\\mathbb C^2\\to \\mathbb C$, $\\pi(\\lambda,w)=\\lambda$, be the projection to the first\ncoordinate.\n\nWe say that a torus $\\Gamma\\subset \\{\\partial \\Delta\\}\\times \\mathbb C$ is a {\\bf graphical torus} if\nfor each $\\lambda\\in\\partial \\Delta$, $C_{\\lambda}:=\\pi^{-1}(\\lambda)\\in \\mathbb C\\backslash \\{0\\}$ is a simple closed curve that\nhas winding number $1$ around $0$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[scale =0.7]\n\\node at (0,-3) {$\\Delta$};\n\n\\draw (0,-2) ellipse (2cm and 0.6cm);\n\\draw (0,0) ellipse (2cm and 0.6cm);\n\n\\draw [very thick] (-2,0) to  [in=-90] (-2.2,0.5) to  [out=85, in=170] (-1.9,1.5) to [out=0, in=90] (-1.6,0) to [out=-90, in=0]\n(-1.9,-1.5) to [out=180, in=-90] (-2,0);\n\n\\node at (-2,2) {$C_{-1}$};\n\n\\draw [very thick] (-2+1.8,0+0.5) to  [out=90, in=-90] (-2.2+1.8,0.5+0.5) to  [out=85, in=220] (-1.9+1.8,1.8+0.5) to [out=0, in=90] (-1.6+1.8,0+0.5) to [out=-90, in=0]\n(-1.9+1.8,-1.5+0.5) to [out=180, in=-90] (-2+1.8,0+0.5);\n\n\\node at (0,2.7) {$C_{\\lambda}$};\n\n\\draw [very thick] (-2+3.6,0) to  [out=90, in=-90] (-2.2+3.6,0.5) to   [out=85, in=170] (-1.9+3.6,1.5) to [out=0, in=90] (-1.6+3.6,0) to [out=-90, in=0]\n(-1.9+3.6,-1.5+0) to [out=180, in=-90] (-2+3.6,0);\n\n\\node at (1.7,2) {$C_{1}$};\n\n\\draw [very thick] (-2+7,0+0.5) to  [out=90, in=-90] (-2.2+7,0.5+0.5) to  [out=85, in=220] (-1.9+7,1.8+0.5) to [out=0, in=90] (-1.6+7,0+0.5) to [out=-90, in=0]\n(-1.9+7,-1.5+0.5) to [out=180, in=-90] (-2+7,0+0.5);\n\n\\node at (5.5,2.5) {$C_{\\lambda}$};\n\n\\draw (4.5,-2.3) to (4.5,2.2) to (5.8,3.7) to (5.8, -0.8) to (4.5,-2.3);\n\n\\node at (6.2, 3) {$\\mathbb{C}$};\n\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Torus $\\Gamma$}\n\\end{figure}\n\nThus, the vertical slices $\\{C_{\\lambda}:\\ \\lambda\\in \\mathbb S^1\\}$ give a foliation of $\\Gamma$. We wish to construct a transverse foliation of $\\Gamma$. We will consider holomorphic functions $g_{\\xi}:\\Delta\\to \\mathbb C$, which extend continuously to $\\bar{\\Delta}$ and such that $\\gamma_{\\xi}:=g_{\\xi}(\\partial \\Delta)\\subset \\Gamma$. We will construct a family of holomorphic disks such that $\\{\\gamma_{\\xi}: \\xi \\in \\mathbb S^1\\}$ form another foliation of the torus $\\Gamma$ that is transverse to the original foliation.\n\n\\subsection{Family of Graphical Tori}\n\nLet $\\{C_{\\lambda}^t: t>0, \\lambda\\in \\partial \\Delta\\}$ be smooth curves, such that\n\\begin{enumerate}\n\\item $C^t_{\\lambda}$ have winding number $1$ around $0$;\n\\item for fixed $\\lambda$, $C^t_{\\lambda}$ form a smooth foliation of $\\mathbb C\\backslash\\{0\\}$;\n\\item there exists $\\epsilon>0$, so that $C^t_{\\lambda}=\\{|w|^2=t\\}$ for $t<\\epsilon$.\n\\end{enumerate}\n\nLet\n$$\\Gamma^t=\\{(\\lambda,w):\\ \\lambda\\in \\partial \\Delta, w\\in C^t_{\\lambda}\\}.$$\nWe set $\\Gamma^0=\\{(\\lambda, 0):\\ \\lambda\\in \\partial \\Delta\\}$. We refer to $\\Gamma^t$, $t\\geq 0$ as smooth family of graphical tori, though for $t=0$ it degenerates to a circle $\\Gamma^0$. The superscript $t$ will be applied to indicate the dependence on the torus $\\Gamma^t$.\n\n\\subsection{Holomorphic Transverse Foliation of a Graphical Torus}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{tikzpicture}[scale =0.7]\n\\node at (0,3) {$\\Gamma$};\n\n\\draw [very thick] (-3,0) to [out=90, in=180] (0,2) to [out=0, in=90] (3,0) to [out=-90, in=0] (0,-2) to [out=180, in=-90] (-3,0);\n\n\\draw [very thick] (-1.2,-0.2) to [out=30, in=150] (1.2,-0.2);\n\\draw [very thick] (-1.5,0) to (-1.2,-0.2) to [out=-30, in=-150] (1.2,-0.2) to (1.5,0);\n\n\\draw [thick] (-2.1,0) to [out=90, in=180] (-1,0.7) to [out=0, in=200] (0,1) to [out=20, in=100] (1.9,0) to [out=-80, in=0] (0,-1) to [out=180, in=0] (-1.1,-0.9) to [out=180, in=-90] (-2.1,0);\n\n\\draw [thick] (-2.6,0) to [out=90, in=180] (-1.3,1) to [out=0, in=200] (0,1.5) to [out=20, in=100] (2.4,0) to [out=-80, in=0] (0,-1.4) to [out=180, in=0] (-1.2,-1.2) to [out=180, in=-90] (-2.6,0);\n\n\n\\draw  (-0.9, -0.35) to [out=190, in=160] (-1.2,-1.84);\n\\draw  [dashed] (-0.9, -0.35) to [out=-10, in=-20] (-1.2,-1.84);\n\\node at (-1.2,-2.3) {$C_\\lambda$};\n\n\\draw [dashed] (0.9, -0.35) to [out=10, in=10] (1.0,-1.86);\n\\draw (0.9, -0.35) to [out=195, in=170] (1.0,-1.86);\n\\node at (1.4,-2.3) {$C_1$};\n\n\\draw [thick] (2.2,0.6)--(3.7,2);\n\\node at (3.9,2) {$\\gamma_\\xi$};\n\n\\draw [thick] (1.85,-0.4)--(3.5,0.5);\n\\node at (3.7,0.5) {$\\gamma_\\eta$};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{Holomorphic Transverse Foliation of the Torus $\\Gamma$}\n\\end{figure}\n\nLet $\\Gamma$ be a graphical torus. Let $g:\\Delta\\to \\mathbb C$ be a holomorphic function that extends continuously to the closure $\\bar{\\Delta}$.\nWe say that the function $g:\\bar{\\Delta}\\to \\mathbb C$ defines a {\\bf holomorphic disk} $D:=\\{(\\lambda,g(\\lambda)):\\lambda \\in \\Delta\\}\\subset \\mathbb C^2$ with a {\\bf trace} $\\gamma:=\\partial D$. \n\n\nWe will construct foliations of graphical tori by traces of holomorphic disks.   To do this, we will require additional properties:\n\nWe say that a function $g:\\bar{\\Delta}\\times \\mathbb S^1\\to\\mathbb C$ defines a {\\bf holomorphic transverse foliation} of a graphical torus\n$\\Gamma$ if\n\\begin{enumerate}\n\\item $g$ is continuous.  \n\\item for each $\\xi\\in \\mathbb S^1$, we let $\\{\\gamma_{\\xi}:=g(\\lambda,\\xi): \\lambda\\in \\partial \\Delta\\}$. The curves $\\gamma_{\\xi}$ are simple, pairwise disjoint and define a foliation of $\\Gamma$.\n\\item Let $g_{\\xi}(\\lambda):=g(\\lambda,\\xi)$, $g_{\\xi}:\\Delta\\to \\mathbb C$ is holomorphic, $g_{\\xi}\\in C^{1,\\alpha}(\\bar{\\Delta})$\n\\item $g_{\\xi}(\\lambda)\\neq 0$, for all $\\xi\\in \\mathbb S^1$, $\\lambda\\in \\Delta$\n\\item $g_{\\xi}(\\lambda)\\neq g_{\\eta}(\\lambda)$, for every $\\lambda\\in \\Delta$ and distinct $\\xi,\\eta\\in \\mathbb S^1$.\n\\end{enumerate}\n\nWe will also consider holomorphic transverse foliations of a smooth family graphical tori $\\{\\Gamma^t\\}$.  This refers to a smooth family of foliations of graphical tori $\\Gamma^t$ with the additional assumption that the disks from $\\Gamma^{t_1}$ are disjoint from the disks from $\\Gamma^{t_2}$ if $t_1\\ne t_2$.\n\nIn fact the leaves in all of our foliations will be closed, and thus they are also fibrations by curves.\n\n\n\n\\section{Holomorphic transverse foliations and the Finite $\\lambda$-lemma}\\label{sec:filling_theorem}\n\n\\newtheorem*{fill_th}{Filling Theorem} \n\n\\begin{fill_th} \\label{te:torus_foliation} Let $\\Gamma$ be a graphical\ntorus, then there exist a function $g:\\bar{\\Delta}\\times \\mathbb\nS^1\\to \\mathbb C$ that defines a holomorphic transverse foliation\nof $\\Gamma$. Moreover, the foliation is unique in the following strong\nsense: if there is a function $h:\\bar{\\Delta}\\to\\mathbb C$ that\ndefines a holomorphic disk with trace in $\\Gamma$, and if $h(\\lambda)\\neq 0$ for\n$\\lambda\\in \\Delta$, then there exists $\\xi\\in \\mathbb S^1$ so\nthat $h=g_{\\xi}$.\n\\end{fill_th}\n\nWe need the following slightly stronger statement to deduce the\nFinite $\\lambda$-lemma:\n\n\\newtheorem*{fill_th'}{Filling Theorem$'$}\n\n\\begin{fill_th'} Let $\\Gamma^t$, $t\\in [0,\\infty)$ be a family of graphical tori. There exists a function\n$g:\\bar{\\Delta}\\times \\mathbb S^1\\times [0,\\infty)\\to \\mathbb C$ that defines a\nholomorphic transverse foliation of the family $\\Gamma^t$. And the foliation is unique in the above mentioned strong sense.\n\\end{fill_th'}\n\nThe reduction of the Finite $\\lambda$-lemma to Filling Theorem$'$\ncan be found in \\cite{Slodkowski}. \n\\begin{proof}[Reduction of the Finite $\\lambda$-lemma to Filling Theorem$'$]\nLet $f$ be a holomorphic motion of the points $a_1,\\dots,a_n$. We\nneed to extend the motion $f$ to one more point $a_{n+1}$. To\nachieve that we construct a holomorphic motion of all of $\\mathbb\nC$ and pick the leaf that passes through the point $a_{n+1}$.\n\nWe normalize the motion so that $a_1=0$, $f(\\lambda,0)=0$ for all\n$\\lambda\\in \\Delta$. Let $\\lambda=re^{i\\theta}$. For each $r\\in\n[0,1)$, $e^{i\\theta}\\in \\mathbb S^1$ the derivative $\\frac{\\partial\nf}{\\partial r} (\\lambda, a_i)$ defines a vector $v_{\\theta}(r,a_i)$ in $\\mathbb C$. We\ncan extend it to a smooth family of vector fields $v_{\\theta}(r,\\cdot)$\non $\\mathbb C$. By integrating the vector field for $r\\in [0,1)$ and taking the union of solutions over\n$\\xi\\in \\mathbb S^1$,\nwe get a smooth motion $g:\\Delta\\times \\mathbb C\\to \\mathbb{C}$ such that\n$g(\\lambda,a_i)=f(\\lambda, a_i)$.\n\nLet $C_0^t$ be a smooth family of simple curves that foliate\n$\\mathbb C\\backslash \\{0\\}$. We choose the foliation so that\ndifferent $a_i$ belong to different curves $C_0^t$. Take $r<1$. Let\n$\\mathbb S_r=\\{\\lambda:\\,|\\lambda|=r\\}$. Let\n$C^{t}_\\lambda=g(\\lambda, C^t_0)$ for $\\lambda \\in \\mathbb S_r$.\n\nBy Filling Theorem$'$, there exists a holomorphic\nmotion with the prescribed traces $\\Gamma^t_r=\\{(\\lambda,\nC^t_{\\lambda}):\\,\\lambda\\in \\mathbb S_r\\}$. By the uniqueness, it\ncoincides with $f$ on points $a_1, \\dots, a_n$. By taking the\nlimit as $r\\to 1$, we obtain a holomorphic motion of $\\mathbb C$\nthat coincides with $f$ on $a_1,\\dots, a_n$.\n\\end{proof}\n\n\\section{Trapping holomorphic disks inside pseudoconvex domains} \\label{sec:trapping_disks}\n\nThe aim of the section is to prove a priori estimates for the derivative of a disk with the trace in a graphical torus (Corollary \\ref{cor:apriori}), which is the heart of our proof of the $\\lambda$-lemma. \n\n\\subsection{Estimates for holomorphic disks trapped inside strictly\npseudoconvex domains}\n\nThe next theorem is from \\cite{BK}, \\cite{BG}. We do not use the result of the theorem. We provide the proof to shed light on the technique we use and put the results in\na general context. \n\n\\begin{theorem} \\cite{BK}, \\cite{BG}\\label{te:angle_estimate} Let $\\Omega$ be a strictly pseudoconvex domain, and let $M$ be a totally real $2$-dimensional manifold,\n$M\\subset \\partial \\Omega$. Let $g:\\Delta\\to \\Omega$ be an injective\nholomorphic function that extends as a $C^1$ smooth function to the closure $\\bar{\\Delta}$. Set\n$D=g(\\Delta)$. Assume that $\\gamma:=\\partial D\\subset M$. Then there\nis a constant $\\alpha = \\alpha(M, \\Omega)$, so that the angle\n$\\angle(T_p\\gamma,\\xi_p)>\\alpha$ is uniformly large, independently of $D$.\n\\end{theorem}\n\n\n\n\\begin{lemma}\\label{lem:transv_ch} Under hypothesis of Theorem \\ref{te:angle_estimate}, for every point $p\\in \\gamma$, $T_p\\gamma$ is transverse to the characteristic field of directions $\\chi(M, \\Omega)$.\n\\end{lemma}\n\n\\begin{proof} Let $\\rho$ be a strictly psh function such that $\\Omega=\\{\\rho<0\\}$. The function $\\rho\\circ g:\\Delta\\to \\mathbb R$ is subharmonic. Let $p\\in \\partial \\Delta$. By the Hopf Lemma, the radial derivative $\\frac{\\partial \\left(\\rho\\circ g\\right)}{\\partial r}(p)>0$. Let $\\xi_p$ be a vector that defines the characteristic direction in a point $p$. The normal vector to the disk $g(\\Delta)$ in a point $p$ is $iT_p\\gamma$. It does not belong to the tangent plane to $\\partial \\Omega,$ so $iT_p\\gamma$ is transverse to $i\\xi_p$. Therefore, $T_p\\gamma$ is transverse to $\\xi_p$.\n\\end{proof}\n\nLet $n_p$ be the unit outward normal vector to the hypersurface $\\partial \\Omega$. The vectors $(\\xi_p, i\\xi_p, n_p, in_p)$ form an\northonormal basis in $\\mathbb C^2\\approx \\mathbb R^4$ with respect to Euclidean inner product $(\\cdot,\\cdot)$. The vectors $in_p$ and $\\xi_p$ form an orthonormal basis for $T_pM$.   Given $\\alpha$, we define a conical neighborhood of $\\xi_p$:\n$$K_{\\alpha}=\\{v\\in T_pM: (v,\\xi_p)>\\alpha(v,in_p)\\} \\subset T_pM.$$\n\n\\begin{lemma}\\label{lem:family_pseudoconvex} Let $\\Omega$ be a strictly pseudoconvex domain, and let $M\\subset \\partial \\Omega$ be totally real.  There\nexist $\\alpha>0$, and a continuous family of strictly\npseudoconvex domains $\\Omega_{\\epsilon}$ such that $M\\subset \\partial\n\\Omega_{\\epsilon}$, and the characteristic fields of directions $\\chi(M,\n\\Omega_{\\epsilon})$ fill the cone-fields $K_{\\alpha}$.\n\\end{lemma}\n\n\\begin{proof} The manifold $M$ separates $\\partial \\Omega$ into two parts\n$(\\partial \\Omega)_1$, $(\\partial \\Omega)_2$. Let $h$ be a smooth\nfunction such that\n\\begin{enumerate}\n\\item $h|_{M}=0$;\n\\item $h|_{(\\partial \\Omega)_1}>0$, $h_{(\\partial \\Omega)_2}<0$;\n\\item $\\frac{\\partial h}{\\partial (i\\xi_p)}>0$, for each $p\\in M$.\n\\end{enumerate}\nLet us denote by $\\vec{n}$ the normal field to the hypersurfaces\n$\\rho=const$. Since we can identify $T_p\\mathbb C^2$ with $\\mathbb\nC^2$, we can treat the normal vector field $n$ as a function\ndefined in a neighborhood of $\\partial \\Omega$. We use the same\nletter $n$ for this function. Let $\\rho_{\\epsilon}(z)=\\rho(z+\\epsilon h\\vec{n})$,\n$\\Omega_{\\epsilon}=\\{\\rho_{\\epsilon}<0\\}$. Then there exists $\\delta$, so that for\n$|\\epsilon|<\\delta$, $\\rho_{\\epsilon}$ are plurisubharmonic. Therefore, $\\Omega_{\\epsilon}$\nare strictly pseudoconvex, and characteristic fields of directions\nto $\\Omega_{\\epsilon}$ fill the cone field $K_{\\alpha}$.\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{te:angle_estimate}]\n\nLet $D\\subset \\Omega$, $\\partial D\\subset \\partial \\Omega$. Then\nby Lemma \\ref{lem:family_pseudoconvex}, there exists a continuous\nfamily of strictly pseudoconvex domains $\\Omega_{\\epsilon}$, $|\\epsilon|<\\delta$\nso that their characteristic fields of directions fill\n$C_{\\alpha}$, for some $\\alpha>0$. By Lemma\n\\ref{lem:disk_trapping}, $D\\subset \\Omega_{\\epsilon}$ for $|\\epsilon|<\\delta$.\nTherefore, an angle estimate follows.\n\\end{proof}\n\n\\subsection{Pseudoconvex domains for Graphical Tori}\nWe wish to obtain the angle estimates for graphical tori. Let $\\eta_p$ be a vector that is tangent to the curve\n$C_{\\lambda}$ in a point $p$. We want to think of $\\eta_p$ as a characteristic direction. However, a priori a graphical torus $\\Gamma$ does not belong\nto a pseudoconvex domain. It belongs to a Levi flat domain $\\{|\\lambda|=1\\}\\times \\mathbb{C}$. Our strategy is to curve this Levi\nflat domain to obtain a family of pseudoconvex domains whose boundaries contain the torus $\\Gamma$ and so that characteristic directions span a wedge around $\\eta_p$. \n\n\\begin{theorem}\\label{te:anlgle_estimate2} Let $\\Gamma$ be a graphical torus. Assume that $g:\\Delta\\to \\mathbb C$ defines a holomorphic disk $D$ with the trace $\\gamma\\subset \\Gamma$, $g(\\lambda)\\neq 0$. Then there exists a constant $\\alpha=\\alpha(\\Gamma)>0$ (independent of $D$) so that the angle $\\angle(\\eta_p,\nT_p\\gamma)$ is bounded below by $\\alpha$ independently of $D$. \n\\end{theorem}\n\nWe need Lemmas \\ref{lem:phi}, \\ref{lem:psi} and \\ref{lem:pseudo} to prove Theorem\n\\ref{te:anlgle_estimate2}.\n\nConsider a family of the graphical tori $\\Gamma^t$, $\\Gamma^1=\\Gamma$. Let $F:\\mathbb S^1\\times \\mathbb C\\to \\mathbb R$ be a defining function, $F^{-1}(t)=\\Gamma^t$. \nLet us extend $F$ to a smooth function $F: \\bar{\\Delta}\\times \\mathbb C\\to \\mathbb R$, so\nthat $F(\\lambda,w)=|w|^2$ for all $\\lambda\\in \\bar{\\Delta}$,\n$|w|\\leq \\epsilon$. We can also satisfy the condition $F'_w\\neq 0$.\n\n\\begin{lemma}\\label{lem:phi} There exists a function\n$\\phi:\\bar{\\Delta}\\times \\mathbb C\\to \\mathbb R_{\\geq 0}$, so that $\\phi$ is\nsmooth, $\\Delta_{w}\\phi>0$, and restriction of $\\phi$ to $\\mathbb S^1\\times \\mathbb C$ defines a foliation of $\\mathbb S^1\\times \\mathbb C$ by $\\Gamma^t$. We also require that for $|\\lambda|=1$\n$\\phi_{\\lambda}^{-1}(1)=C_{\\lambda}.$\n\\end{lemma}\n\n\\begin{proof} Let $F(\\lambda, w)$ be the extension defined earlier. Let $\\rho:\\mathbb R_+\\to \\mathbb R_+$\nbe an increasing convex function, $\\rho(0)=0,$ $\\rho(1)=1$. Then $\\phi=\\rho\\circ F$ is also an extension of a defining function of the foliation as well.\n\n\\begin{equation}\\label{eq:1}\n\\Delta_w(\\rho\\circ F)=\\frac14\\rho''|F_w|^2+\\frac14\\rho'\\Delta_w F\n\\end{equation}\n\nSince $F'_w(\\lambda, w)\\neq 0$, when $w\\neq 0$, so that $\\Delta_w(\\rho\\circ F)>0$ away from a neighborhood of $w=0$.  In a neighborhood of $0$, $\\Delta_w F=4$. By\ntaking $\\rho'(0)>0$, one can insure that $\\Delta(\\rho\\circ F)>0$.\n\nLet us set $\\phi=\\rho\\circ F$, then $\\phi_{\\lambda}^{-1}=C_{\\lambda}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:psi} There exists a function $\\psi:\\bar{\\Delta}\\times \\mathbb C\\to \\mathbb R\\cup\\{-\\infty\\}$, so that $\\psi$ is smooth, $\\Delta_{w} \\psi<0$, and restriction of $\\psi$ to $\\mathbb S^1\\times \\mathbb C$ defines a foliation of $\\mathbb S^1\\times \\mathbb C$ by $\\Gamma^t$. We require that $\\psi(\\lambda,0)=-\\infty$ for all $\\lambda\\in \\bar{\\Delta}$. We also require that for $|\\lambda|=1$, $\\psi_{\\lambda}^{-1}(t)=C_{\\lambda}$.\n\\end{lemma}\n\n\\begin{proof}\nConsider a function $\\psi=c \\rho\\circ \\ln F$, where $\\rho$ is increasing, concave function, $\\rho(-\\infty)=-\\infty$.\n\n$$\\Delta_w(\\rho\\circ\\ln F)=\\frac14\\rho''\\frac{|F_w|^2}{F^2}+\\frac14\\rho' \\Delta_w (\\ln F)$$\n\nSince $F'_w\\neq 0$ when $w\\neq 0$, we can make $\\Delta_w(\\rho\\circ \\ln F)<0$. In a neighborhood of $w=0$, $\\Delta_w(\\ln F)=0$, therefore \n$\\Delta_w (\\rho\\circ \\ln F)<0$. By choosing a constant $c$, we can ensure that $\\psi_{\\lambda}^{-1}(1)=C_{\\lambda}$.\n\n\\end{proof}\n\nLet $T\\Gamma$ be the tangent space of the graphical torus $\\Gamma$. Let $K_{\\alpha}\\subset T\\Gamma$ be the cone field:\n\n$$K_{\\alpha}:=\\{(p, v):\\, v\\in T_pT, (v,\\eta_p)>\\alpha(v, \\frac{\\partial }{\\partial \\theta})\\}.$$\n$$K^{\\circ}_{\\alpha}:=\\{(p,v)\\in K_{\\alpha}:\\, v\\neq c \\eta_p, c\\in \\mathbb R \\}$$\n\n\\begin{lemma} \\label{lem:pseudo} For a graphical torus $\\Gamma$, there exist a family of pseudoconvex domains $\\Omega_{\\epsilon}$, $\\epsilon\\in [-\\delta, 0)\\cup(0,\\delta]$ and $\\alpha>0$, so that $\\Gamma\\subset \\partial \\Omega_{\\epsilon}$ and characteristic directions $\\chi(T,\\Omega_{\\epsilon})$ fill $K^{\\circ}_{\\alpha}$.\n\\end{lemma}\n\n\\begin{proof} Take\n$$\\omega_{\\epsilon}:=\\frac{1}{\\epsilon}(|\\lambda|^2-1)+\\phi,$$\nwhere $\\phi$ is a function constructed in Lemma \\ref{lem:phi}.\n\n$$\\mbox{Hess}\\,\\omega_{\\epsilon}=\\left( \\begin{array}{ll}\\frac{1}{\\epsilon}+\\frac{\\partial^2\\phi}{\\partial \\lambda\\partial\n\\overline{\\lambda}} & \\frac{\\partial^2\\phi}{\\partial w\\partial\\overline{\\lambda}}\\\\ \\frac{\\partial^2\\phi}{\\partial\\overline{w}\\partial\\lambda} & \\Delta_w\\phi\\end{array}\\right)$$\n\nFor small enough $\\epsilon$, the Hessian is positive definite, so the function $\\omega_{\\epsilon}$ is strictly plurisubharmonic.\nThe domains\n$$\\Omega_{\\epsilon}=\\{(\\lambda, w):\\,\\omega_{\\epsilon}(\\lambda, w)< 1\\}.$$\nare strictly pseudoconvex for small $\\epsilon$.\n\nLet $D$ be a holomorphic disk with the trace in $\\Gamma$.\nThe domains $\\Omega_{\\epsilon}$ converge to\n$|\\lambda|<1$.  Therefore, by Lemma~\\ref{lem:disk_trapping}, the disk $D$ is trapped in\n$\\Omega_{\\epsilon}$ for all small enough $\\epsilon$.\n\nFor small $\\epsilon$, the function\n$$\\sigma_{\\epsilon}(\\lambda,w):=\\frac{1}{\\epsilon}(|\\lambda|^2-1)-\\psi$$\nis strictly plurisubharmonic.  By the same reasoning, the disks are trapped in\n$$\\Sigma_{\\epsilon}=\\{(\\lambda, w):\\, \\sigma_{\\epsilon}<-1\\}$$\nwhen $\\epsilon$ is sufficiently small.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{te:anlgle_estimate2}]\nBy Lemma \\ref{lem:transv_ch}, the tangent $T_p\\gamma$ is transverse to characteristic directions. Therefore, the angle estimate follows.\n\\end{proof}\n\n\\begin{corollary} \\label{cor:apriori} Let $g:\\Delta\\to \\mathbb C$ define a holomorphic disk with the trace in $\\Gamma$, $g(\\lambda)\\neq 0$ for $\\lambda\\in \\Delta$. Assume that  $g\\in C^1(\\bar{\\Delta}).$  Then there exists $C$ depending only on $\\Gamma$ such\nthat $|g'(\\lambda)|<C$ for all $\\lambda\\in \\bar{\\Delta}$. The derivative estimate stays valid for graphical tori that are\nsmall perturbations of $\\Gamma$.\n\\end{corollary}\n\n\\begin{proof} It is enough to estimate $g'(\\lambda)$ for $|\\lambda|=1$. Then $\\lambda=e^{i\\theta}$,\nso $|g'_{\\lambda}|=|g'_{\\theta}|$. Let $u,v,\\theta,r$ be an\northornormal system of coordinates in a neighborhood of $\\Gamma$. We\nassume that $u|_{\\Gamma}$ is a coordinate along $C_{\\lambda}$ and $v,u$\nare coordinates in $\\lambda=const$ plane. Then\n$g'_{\\theta}=u'_{\\theta}$ and the angle estimate implies that\n$|u'_{\\theta}|$ is uniformly bounded from below.\n\\end{proof}\n\n\n\n\\section{Proof of the Filling Theorem}\\label{sec:proof}\n\nThe proof is by continuity method. At many points we follow the treatment of \\cite{AM}. For each $\\lambda\\in \\mathbb\nS^1$ we can foliate interior of $C^t_{\\lambda}\\backslash\n\\{0\\}$ by simple smooth curves $C_{\\lambda}^s$, $s\\in (0,t)$ so that\n\\begin{enumerate}\n\\item $C^s_{\\lambda}=\\{|z|=s\\}$ for $s\\leq \\epsilon$;\n\\item $C^s_{\\lambda}$ depend smoothly on $\\lambda$.\n\\end{enumerate}\n\nLet\n$$\\Gamma^t=\\{(\\lambda,w):\\, \\lambda\\in \\mathbb S^1, w\\in\nC^t_{\\lambda}\\}$$\n$$\\Gamma^0=\\{(\\lambda, 0):\\, \\lambda\\in \\mathbb S^1\\}$$\n\n$\\Gamma^t$ by definition is a smooth family of graphical tori, ${\\Gamma}^1=\\Gamma$.\nFor $t\\leq \\epsilon$, the tori $\\Gamma^t$ are foliated by the vertical\nleaves $w=\\mbox{const}$. We will prove that the set $S$ of\nparameters $t$ such that $\\Gamma^t$ is foliated is open and closed in\n$[0,1]$, so $S=[0,1]$, and the torus $\\Gamma$ is foliated. Moreover, we\nwill prove that the foliation is unique in the strong sense.\n\nLet $F:\\mathbb S^1\\times \\mathbb C\\to \\mathbb R$ be a\ndefining function of the foliations $C^t_{\\lambda}$. For each\nfixed $\\lambda$,\n$$C^t_{\\lambda}=\\{(\\lambda, w): F(\\lambda, w)=t\\}.$$\nThe function $F$ depends smoothly on $\\lambda$. We assume that\n$F'_{w}(\\lambda, w)\\neq 0$ for $w\\neq 0$, $\\lambda\\in \\mathbb\nS^1.$\n\n\\begin{lemma} \\label{lem:winding_number} Assume that the winding\nnumber of a curve $\\{\\gamma(\\lambda):\\lambda\\in \\mathbb\nS^1_{\\lambda}\\}$ around $0$ is equal to zero. Then the winding\nnumber of the curve $\\{F'_w(\\lambda,\n\\gamma(\\lambda)):\\,\\lambda\\in\\mathbb S^1\\}$ around $0$\nis equal to zero.\n\\end{lemma}\n\\begin{proof} There is a homotopy of the curve $\\gamma,$ $G:\\gamma\\times [0,1]\\to \\mathbb C\\backslash\\{0\\}$ so that $G(\\gamma\\times\\{0\\})=\\gamma$, $G(\\gamma\\times\\{1\\})=\\mbox{const}$. The winding\nnumber of the curves $\\{F'_w(\\lambda, \\gamma^t(\\lambda)):\\, \\lambda\n\\in \\mathbb S^1\\}$ around $0$ is well defined, so it stays constant. Hence, it\nis equal to zero.\n\\end{proof}\n\n\\subsection{Regularity} \n\n\\begin{theorem}\\label{te:regularity} Let $\\Gamma$ be a graphical torus. Let $g:\\bar{\\Delta}\\to \\mathbb C$ be a function that defines a holomorphic disk with the trace $g(\\partial \\Delta) \\in \\Gamma.$ Assume $g'\\in L^{\\infty}(\\Delta)$, $g\\neq 0$ $\\forall \\lambda\\in \\Delta$. Then $g\\in C^{1,\\alpha}(\\bar{\\Delta})$, $0<\\alpha<1$.\n\\end{theorem}\n\\begin{proof} We include $\\Gamma$ into a family of graphical tori $\\Gamma^t$ with $\\Gamma^1=\\Gamma$. Let $F:\\mathbb S^1\\times \\mathbb C\\to \\mathbb R$ be a defining function for $\\Gamma^t$, $F^{-1}(t)=\\Gamma^t$. \nSince the trace of $g$ is in $\\Gamma$ we have equation:\n\n\\begin{equation}\\label{eq:def}\nF(\\lambda,g(\\lambda))=1. \n\\end{equation}\n\nLet $\\lambda=e^{i\\theta}.$ Since $g'\\in L^{\\infty}(\\Delta)$, the bounded radial limits exist almost everywhere. The function $g$ extends to be $C^{\\alpha}$ on the closed disk, and the partial derivative $g_{\\theta}$ exist a.e. We differentiate equation (\\ref{eq:def}) a.e. with respect to $\\theta$ and obtain: \n\\begin{equation}\\label{eq:der}\\lambda i F_{\\lambda}(\\lambda,g(\\lambda))-\\mbox{Im}\\left(F_w(\\lambda,g(\\lambda))g'(\\lambda)\\lambda\\right)=0.\\end{equation}\n\nThe winding number of $\\{g(\\lambda):\\lambda\\in \\mathbb S^1\\}$ around $0$ is zero, and by Lemma \\ref{lem:winding_number}, the winding number of $\\{F_w(\\lambda,g(\\lambda)):\\,\\lambda \\in\\mathbb S^1\\}$ around $0$ is zero as well. Thus we can take the logarithm and obtain \n$$F_w(\\lambda, g(\\lambda))=e^{a(\\lambda)+ib(\\lambda)}.$$\nThe left hand-side is $\\alpha$-H\\\"{o}lder continuous, so $b(\\lambda)$ is $\\alpha$-H\\\"{o}lder continuous function, and so is its Hilbert transform $Hb(\\lambda)$. Thus equation (\\ref{eq:der}) becomes\n$$\\mbox{Im} \\left(\\lambda e^{Hb(\\lambda)-ib(\\lambda)} g'(\\lambda)\\right)=e^{-a(\\lambda)}F'_{\\lambda}(\\lambda, g(\\lambda))\\lambda$$ for almost every $\\theta$. Since the right hand side is $C^{\\alpha}$, so is the\nleft hand side. Further the left hand side is the imaginary part of an analytic function \nso the function  \n$\\lambda e^{Hb(\\lambda)-ib(\\lambda)} g'(\\lambda)$ itself is $C^{\\alpha}$. Therefore, $g'\\in C^{\\alpha}(\\bar{\\Delta})$.\n\\end{proof}\n\n\\subsection{Openness}\n\nIn \\cite{B}, the stability of foliation by holomorphic disks is proved if one starts from the standard torus.\n\n\\begin{theorem}\\label{te:open} Let $\\Gamma^t$ be a family of graphical tori, $t\\in\n[0,\\infty)$. Assume that a function $g^{t_0}:\\bar{\\Delta}\\times \\mathbb\nS^1\\to \\mathbb C$ defines a holomorphic transverse foliation of a\ngraphical torus $\\Gamma^{t_0}$. Then there exists $\\delta$ and\na function $\\tilde{g}:\\bar{\\Delta}\\times \\mathbb S^1\\times\n(t_0-\\delta,t_0+\\delta)\\to \\mathbb C$ that defines a transverse\nholomorphic foliation of $\\Gamma^t$ for $|t-t_0|<\\delta$.\n\\end{theorem}\n\n\\begin{proof}\nHilbert transform\n$$H:C^{1,\\alpha}(\\mathbb S^1)\\to C^{1,\\alpha}(\\mathbb S^1)$$\nis a bounded linear operator. We change the standard normalization $Hu(0)=0$ to $Hu(1)=0$. We denote by $C_{\\mathbb R}^{1,\\alpha}(\\mathbb S^1)\\subset C^{1,\\alpha}(\\mathbb S^1)$ be the subspace of real-valued functions. The curve\n$\\{g^{t_0}_{\\xi}(\\lambda), \\lambda\\in \\mathbb S^1\\}$\nhas winding number $0$ around zero, since\n$g^{t_0}_{\\xi}(\\lambda)\\neq 0$ for $\\lambda\\in \\Delta$. Therefore, by Lemma \\ref{lem:winding_number}, the curve\n$\\{F_w(\\lambda,g^{t_0}_{\\xi}(\\lambda)):\\, \\lambda \\in \\mathbb S^1\\}$ has winding number $0$\naround $0$:\n$$F_w(\\lambda,g_{\\xi}^{t_0}(\\lambda))=e^{a_{\\xi}(\\lambda)+ib_{\\xi}(\\lambda)},$$\nwhere $\\alpha_{\\xi}(\\lambda)$, $b_{\\xi}(\\lambda)$ are H\\\"{o}lder continuous\nwith exponent $\\alpha$. Thus, $Hb_{\\xi}(\\lambda)$ is H\\\"{o}lder\ncontinuous as well.\n\n$X_{\\xi}(\\lambda):=e^{Hb_{\\xi}(\\lambda)-ib_{\\xi}(\\lambda)}$ is a holomorphic\nfunction on $\\Delta$ and is proportional to the normal vector to\n$C^t_{\\lambda}$ in points $(\\lambda,g^t_{\\xi}(\\lambda))$.\n\nFunctions of the form $(u(\\lambda)+iHu(\\lambda))X_{\\xi}(\\lambda)$ give\nall holomorphic functions that are H\\\"{o}lder continuous up to the\nboundary with the condition that $(u_{\\xi}(1)+i Hu_{\\xi}(1))X_{\\xi}(1)$ is proportional\nto the normal vector to $C_1^t$ in a point $g^t_{\\xi}(1)$. There\nexists an $\\epsilon$ such that for each point $\\eta\\in C^t_1$,\n$|t-t_0|<\\epsilon$, there is only one normal vector that\nintersects $C_1^t$ in a point $\\eta$.\n\nThe space $C^0(\\mathbb S^1, C^{1,\\alpha}(\\mathbb\nS^1))$ is a Banach space with the norm\n$$||u||=\\sup_{\\xi\\in \\mathbb S^1, \\lambda\\in \\mathbb S^1} |u_{\\xi}(\\lambda)|+\\sup_{\\xi\\in \\mathbb S^1, \\lambda\\in \\mathbb S^1} |u'_{\\xi}(\\lambda)|+\\sup_{\\xi\\in \\mathbb S^1,\\lambda_1\\neq \\lambda_2\\in \\mathbb S^1}\\frac{|u'_\\xi(\\lambda_1)-u'_{\\xi}(\\lambda_2)|}{|\\lambda_1-\\lambda_2|^{\\alpha}}.$$\nConsider an operator \n$${\\mathcal F}:\\mathbb R_t\\times C^0(\\mathbb S^1_{\\xi},\nC_{\\mathbb R}^{1,\\alpha}(\\mathbb S^1_{\\lambda}))\\to C^0(\\mathbb S_{\\xi}^1,\nC_{\\mathbb R}^{1,\\alpha}(\\mathbb S_{\\lambda}^1)):$$\nwhere ${\\mathcal F}$ is a function of two variable $(t,u_{\\xi})$. We consider function $u_\\xi(\\lambda)$ as an element of $C^0({\\Bbb S}_\\xi^1,C^{1,\\alpha}({\\Bbb S}_\\lambda^1)).$ \n$${\\mathcal F}(t,u): {\\Bbb S}^1\\ni (t,\\xi)\\to F \\left(  \\lambda, g_\\xi(\\lambda) + (u_\\xi + i H u_\\xi)X_\\xi(\\lambda) \\right)   - t  \\in C^{1,\\alpha}({\\Bbb S}_\\lambda^1) $$\nFor $0<\\alpha<1$, $H$ is a bounded linear operator, so ${\\mathcal F}$ is a continuous mapping of Banach spaces.  Further, when ${\\mathcal F}$ is considered as a \nmap from ${\\Bbb R}\\times {\\Bbb S}^1_\\xi$ to $C^{0,\\alpha}({\\Bbb S}^1_\\lambda)$, it is  differentiable, and we compute the differential of ${\\mathcal F}$ at $u_\\xi=0$ in the direction $\\delta u_\\xi$: \n$$D{\\mathcal F}(t,0; \\delta u_\\xi) = e^{a_\\xi(\\lambda) - Hb_\\xi(\\lambda)}\\delta u_\\xi(\\lambda).$$\nSince  ${\\mathcal F}(t,0; \\delta u_\\xi)$ is an invertible linear operator, we can define $u^t_{\\xi}$ as the unique element of $C^0({\\Bbb S}_\\xi^1,C^{1,\\alpha}({\\Bbb S}_\\lambda^1))$ satisfying\n${\\mathcal F}(t, u_{\\xi}^t)=0.$ \nAnd the function $\\tilde{g}(\\lambda, \\xi,t)=g^{t_0}_{\\xi}(\\lambda)+u_{\\xi}^t(\\lambda)$ defines a\nholomorphic transverse foliation and is of class $C^{1,\\alpha}$ on\n$\\bar{\\Delta}$. By continuity, for $\\xi\\neq \\eta$, $g^t_{\\xi}(\\lambda)\\neq g^{t}_{\\eta}(\\lambda)$ for $\\lambda\\in \\Delta$.\n\n\\end{proof}\nThis also gives us the openness for one disk.\n\n\\begin{theorem}\\label{te:open_leaf} Let $\\Gamma^t$, $t\\in I$ be a family of graphical tori. Let $g^{t_0}:\\bar{\\Delta}\\to \\mathbb C$ be a function that defines a holomorphic disk with the trace in the torus $\\Gamma^{t_0}$. Assume that $g^{t_0}\\in C^{1,\\alpha}(\\bar{\\Delta})$, \n$g^{t_0}(\\lambda)\\neq 0$ for $\\lambda\\in \\Delta$. \nThen there exists $\\delta$ and a continuous function $g:\\bar{\\Delta}\\times(t_0-\\delta,t_0+\\delta)\\to \\mathbb C$ such that $g^t(\\lambda):=g(\\lambda,t)$ defines a holomorphic disk with the trace in $\\Gamma^t$ and $g^t\\in C^{1,\\alpha}(\\bar{\\Delta})$, $g^t(\\lambda)\\neq 0$ for $\\lambda\\in \\Delta$.\n\\end{theorem}\n\n\\subsection{Closedness}\n\n\\begin{theorem} Let $\\Gamma^t$, $t\\in [0, \\infty)$ be a family of graphical tori.\nSuppose that there exists $g:\\bar{\\Delta}\\times \\mathbb S^1\\times\n[0,t_0)\\to \\mathbb C$ that defines a holomorphic transverse\nfoliation of $\\Gamma^t$. Then $g$ can be extended to\n$g:\\bar{\\Delta}\\times \\mathbb S^1\\times [0,t_0]\\to \\mathbb C$ that\ndefines a holomorphic transverse foliation.\n\\end{theorem}\n\n\\begin{proof} By Corollary \\ref{cor:apriori}, there exists $C$ that depends only on $\\Gamma^{t_0}$ so that $|\\left(g_{\\xi}^t\\right)'|<C$ for $t<t_0$ close to $t_0$. Since the space of bounded holomorphic\nfunctions on $\\Delta$ is compact, we can pass to the limit. Let $g_{\\xi}^{t_0}$ be the limits, $|\\left(g^{t_0}_{\\xi}\\right)'|\\leq C$. By Regularity Theorem \\ref{te:regularity}, \n$g_{\\xi}^{t_0}\\in C^{1,\\alpha}(\\bar{\\Delta})$.\n\\end{proof}\n\nThis also give us closedness for a family of disks.\n\n\\begin{theorem}\\label{te:close_leaf} Let $\\Gamma^t$, $t\\in [0, \\infty)$ be a family of graphical tori.\nAssume that $g:\\bar{\\Delta}\\times [0,t_0)\\to \\mathbb C$ is a\ncontinuous function such that $g^t(\\lambda):=g(\\lambda,t)$ defines\na holomorphic disk with the trace in $\\Gamma^t$, $g^t\\in C^{1,\\alpha}(\\bar{\\Delta})$, $g^t(\\lambda)\\neq 0$ for $\\lambda\\in \\Delta$. Then $g$ can be extended\nto a continuous $g:\\bar{\\Delta}\\times [0,t_0]\\to \\mathbb C$ such\nthat $g^{t_0}$ defines a holomorphic disk with the trace in $\\Gamma^{t_0}$, $g^{t_0}(\\lambda)\\neq 0$ for $\\lambda\\in \\Delta$.\n\\end{theorem}\n\n\n\n\\subsection{Uniqueness} \n\nLet $\\Gamma^c=\\{(\\lambda, w):\\, |w|=c, |\\lambda|=1\\}$ be standard tori.\nLet $g:\\Delta\\to \\mathbb C$ be a function that defines a holomorphic disk with the trace of $g(\\partial \\Delta)\\in \\Gamma^c$. By the Minimum Modulus \nTheorem, $\\min$ of $g$ is attained on the boundary.\nMaximum modulus is attained on the boundary as well. So\n$|g(\\lambda)|=\\mbox{const}$. Therefore, $g(\\lambda)=\\mbox{const}$.\n\n\\begin{theorem}\\label{te:loc_uniqueness} Let $\\Gamma$ be a graphical torus. Let $g,h:\\bar{\\Delta}\\to \\mathbb\nC$ be functions that define holomorphic disks with traces in $\\Gamma$, $g(\\lambda)\\neq 0$, $h(\\lambda)\\neq 0$ for $\\lambda \\in \\Delta$.\nAssume that $g(1)=h(1)$. Then there exists $\\epsilon$ such that $|g(\\lambda)-h(\\lambda)|<\\epsilon\\, \\mbox{for}\\, \\lambda\\in \\mathbb\nS^1$ implies $g(\\lambda)\\equiv h(\\lambda)$.\n\nNote that the same $\\epsilon$ works for tori close to $\\Gamma$.\n\\end{theorem}\n\n\\begin{proof}\n\nLet $a(\\lambda,s)=F(\\lambda,g(\\lambda)+s(h(\\lambda)-g(\\lambda)))$,\n$a(\\lambda,0)=a(\\lambda, 1)=t$. Then\n\\begin{equation}\\label{eq:nrh}\n\\int_0^1 a_s ds=0=Re(h(\\lambda)-g(\\lambda))\\int_0^1 F_w\n\\big(g(\\lambda)+s(h(\\lambda)-g(\\lambda))\\big)ds.\n\\end{equation}\n\nThe winding number of the curve $\\{g(\\lambda):\\lambda\\in \\mathbb\nS^1\\}$ around $0$ is equal to zero. Hence, by Lemma\n\\ref{lem:winding_number}, the winding number of \n$$\n\\{F_w(\\lambda, g(\\lambda)):\\,\\lambda \\in \\mathbb S^1\\}\n$$\naround $0$ is\nequal to zero.\n\nTherefore, for small enough $\\epsilon$, the winding number of the curve\n$$\\left\\{\\int_0^1 F_w\\left(g(\\lambda)+s(h(\\lambda)-g(\\lambda)\\right)ds:\\,\\lambda \\in \\mathbb S^1 \\right\\}$$ around $0$ is equal to zero, so\n\n$$\n\\int_0^1 F_w\\big(g(\\lambda)+s(h(\\lambda)-g(\\lambda))\\big)ds=e^{a(\\lambda)+ib(\\lambda)}.$$\n\nThe function $b(\\lambda)$ is a bounded H\\\"{o}lder continuous function.\n\nBy equation (\\ref{eq:nrh}), $\\arg \\left(g(\\lambda)-h(\\lambda)\\right)=\\frac{\\pi}{2}-b(\\lambda)$, where $b(\\lambda)$ is a bounded function. It contradicts the fact that $g(1)-h(1)=0$. \n\\end{proof}\n\n\\subsection{Global Uniqueness}\n\\begin{theorem}\\label{te:uniqueness} Let $\\Gamma$ be a graphical torus. Let $g^1,h^1:\\bar{\\Delta}\\to \\mathbb\nC$ be functions that define holomorphic disks with traces in $\\Gamma$, $g^1, h^1\\in C^{1,\\alpha}(\\bar{\\Delta})$.\nAssume that $g^1(1)=h^1(1)$. Then $g^1(\\lambda)=h^1(\\lambda)$.\n\\end{theorem}\n\n\\begin{proof}\nWe include torus $\\Gamma$ into a family of\ngraphical tori $\\Gamma^t$, $t\\in [0,1]$, $\\Gamma^1=\\Gamma$. By Theorems\n\\ref{te:open_leaf}, \\ref{te:close_leaf}, there exist functions\n$g,h:\\bar{\\Delta}\\times[0,1]\\to \\mathbb C$ such that $g(\\lambda, 1)=g^1$, $h(\\lambda,1)=h^1$ and\n$g^t(\\lambda):=g(\\lambda, t)$ define holomorphic disks with the traces in tori $\\Gamma^t$, . There\nexists $\\epsilon$ such that for $t<\\epsilon$, $\\Gamma^t=\\{(\\lambda, w):\\, |w|=t, \\lambda \\in \\mathbb S^1_{\\lambda}\\}$, $t\\in [0,1]$ are standard tori with uniqueness of solutions. For $t<\\epsilon$, $g^t\\equiv h^t$. Let $t_0=\\sup \\{t:\\, h^t\\equiv g^t\\}$. If $t_0\\neq 1$, then by applying Theorem \\ref{te:loc_uniqueness}, we get a contradiction.\n\\end{proof}\n\nAt this point we have proved the Filling theorem. For Filling Theorem$'$ the only statement remains is to show that disks for $\\Gamma^{t_1}$ are disjoint from disks\nfor $\\Gamma^{t_2}$ when $t_1\\neq t_2$. Suppose $D^{t_j}$ is a disk with boundary in $\\Gamma^{t_j}$. If $D^{t_1}\\cap D^{t_2}\\neq \\emptyset$, then since the traces are in \n$\\Gamma^{t_1}$ and $\\Gamma^{t_2}$ we will have \n$D_{\\xi_1}^{t_1}\\cap D_{\\xi_2}^{t_1}\\neq \\emptyset$ for all $\\xi_1,\\xi_2\\in \\mathbb S^1$. By Filling Theorem, the disks $D_{\\xi_1}^{t_1}\\cap D_{\\xi_2}^{t_1}=\\emptyset$ for $\\xi_1\\neq\\xi_2$. However, there is a continuous family of disks $D_{\\xi}^t$, $t\\in [t_1,t_2]$, which is a contradiction.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:aa}\n\\input{aa}\n\n\\paragraph*{Outline}\nIn \\cref{section:python}, we apply Grigore's reduction to Python type hints and deduce that they are Turing complete.\n\\Cref{section:real} introduces an alternative design for subtyping machines that simulate the TMs in real time, i.e., make $O(1)$ operations for each TM transition.\nFinally, \\cref{section:implementation} presents our Python implementations of Grigore's original reduction and the new reduction and compares their performances.\n\n\\section{Python Subtyping is Undecidable}\n\\label{section:python}\n\\input{python}\n\n\\section{Real-Time Subtyping Machines}\n\\label{section:real}\n\\input{real}\n\n\\section{Implementation and Performance Experiment}\n\\label{section:implementation}\n\\input{implementation}\n\n\\newpage\n\n\\bibliographystyle{plain}\n\n\\subsection{Nominal Subtyping with Variance}\n\nSubtyping is a type system decision problem.\nGiven types $t$ and $s$, the type system should decide whether type $t$\nis a subtype of $s$, \\[\n    t \\subtype s,\n\\] meaning that every $t$ object is also a member of $s$.\nFor example, every string is an object, $\\text{\\lstinline\/str\/}\\subtype \\text{\\lstinline\/object\/}$, but not every object is a string, $\\text{\\lstinline\/str\/}\\nsubtype \\text{\\lstinline\/object\/}$.\nSubtyping is found, e.g., in variable assignments:\n\\begin{lstlisting}[language=python]\nx: t = ...\ny: s = x  # t$\\color{comment}\\subtype$s\n\\end{lstlisting}\nThe second assignment compiles if and only if $t \\subtype s$.\n\nObject-oriented languages such as Scala, C\\#, and Python use \\emph{nominal subtyping with variance} \\cite{Kennedy:2007}.\n\\emph{Nominal} means that subtyping is guided by inheritance.\nType \\inline{c1} is a subtype of type \\inline{c2} if class \\inline{c1} is a descendant of \\inline{c2}:\n\\begin{lstlisting}[language=python]\nclass s: ...\nclass t(s): ...\nx: t = ...\ny: s = x  # vvv\n\\end{lstlisting}\n\\emph{Variance} enables variability in type arguments.\nBy default, type \\inline{t[v]} is a subtype of \\inline{t[w]} if and only if $\\text{\\inline{v}}=\\text{\\inline{w}}$.\nIf \\inline{t}'s type parameter is covariant, then \\inline{v} can be any subtype of \\inline{w}, $\\text{\\inline{v}}\\subtype\\text{\\inline{w}}$;\nIf it is contravariant, then $\\text{\\inline{v}}\\suptype\\text{\\inline{w}}$.\nIn Python, variance is specified in an argument to \\inline{TypeVar} (the constructor of type parameters).\nFor example, the following Python program uses contravariance and is correctly typed:\n\\begin{lstlisting}\nX = TypeVar(\"X\", contravariant=True)\nclass t(Generic[X]): ...\nx: t[int] = t[object]()  # vvv (int$\\color{comment}\\subtype$object)\n\\end{lstlisting}\n\nKennedy and Pierce showed that nominal subtyping with variance is undecidable \\cite{Kennedy:2007} using a reduction from the Post correspondence problem (PCP) \\cite{Post:1946}.\nTheir work focused on three subtyping features (characteristics):\n\\begin{enumerate}\n    \\item \\emph{Contravariance}:\n        The presence of contravariant type parameters, as described above.\n    \\item \\emph{Expansively recursive inheritance}:\n        The closure of types under the inheritance relation and type decomposition ($\\text{\\inline{t[v]}} \\rightarrow \\text{\\inline{v}}$) is unbounded.\n        (The concept of expansive inheritance was introduced by Viroli using a different definition \\cite{Viroli:2000}.)\n    \\item \\emph{Multiple instantiation inheritance}:\n        Class \\inline{t} is allowed to derive class \\inline{s[$\\cdot$]} multiple times using different type arguments:\n\\begin{lstlisting}[language=python]\nclass t(s[object], s[int]): ...  # not legal in Python\n\\end{lstlisting}\n\\end{enumerate}\nKennedy and Pierce proved that subtyping becomes decidable when contravariance or expansive inheritance are removed, but they were uncertain about the contribution of multiple instantiation inheritance to undecidability.\n\n\\subsection{Subtyping Machines}\n\nTen years later, Grigore showed that Java subtyping is Turing complete using a direct reduction from TMs \\cite{Grigore:2017}.\nThis reduction uses a subset of Java that conforms to Kennedy and Pierce's nominal subtyping with variance.\nGrigore's result settles Kennedy and Pierce's open problem since Java does not support multiple instantiation inheritance \\cite[\\S8.1.5]{JLS:2015}.\n(Intuitively, multiple instantiation inheritance corresponds to non-deterministic subtyping \\cite{Kennedy:2007,Roth:2021}, so while it was useful in the PCP reduction, it is redundant in the TM reduction because deterministic TMs are as expressive as non-deterministic TMs.)\n\nGrigore's reduction can be described as a function\n\\begin{equation}\\label{eq:encoding:function}\n    e(M, w) = \\Delta \\cdot q\n\\end{equation}\nthat encodes TM $M$ and input word $w$ as program $P=\\Delta \\cdot q$ consisting of class table $\\Delta$ and subtyping query $q$.\nThe encoding ensures that TM $M$ accepts $w$ if and only if query $q$ type-checks against $\\Delta$.\n\nThe idea behind the reduction is to encode the TM configuration in the subtyping query $q$.\nRecall that the configuration of TM $M$ comprises \\1 the content of the memory tape, \\2 the location of the machine head on the tape, and \\3 the current state of the machine's finite control.\nFor example, \\cref{figure:tm:configuration} illustrates the initial configuration of $M$:\nThe input word $w=a_1a_2\\ldots a_m$ is written on the tape, the machine head points to the first letter $a_1$, and the current state is the initial state $q_I$.\n\n\\begin{figure}[ht]\n  \\vspace{1ex}\n  \\centering\n  \\begin{tikzpicture}\n    \\tikzstyle{cell}=[draw,minimum width=5ex,minimum height=3.5ex,text=#1]\n    \\tikzstyle{bcell}=[cell=botcolor]\n    \\tikzstyle{scell}=[cell=symbcolor]\n    \\node[text=botcolor] (dots1) {$\\cdots$};\n    \\node[bcell] (b1) [right=0pt of dots1] {$\\bot$};\n    \\node[bcell] (b2) [right=-.5pt of b1] {$\\bot$};\n    \\node[scell] (a1) [right=-.5pt of b2] {$a_1$};\n    \\node[scell] (a2) [right=-.5pt of a1] {$a_2$};\n    \\node[text=symbcolor] (dots2) [right=0pt of a2] {$\\cdots$};\n    \\node[scell] (am) [right=0pt of dots2] {$a_m$};\n    \\node[bcell] (b3) [right=-.5pt of am] {$\\bot$};\n    \\node[bcell] (b4) [right=-.5pt of b3] {$\\bot$};\n    \\node[text=botcolor] (dots3) [right=0pt of b4] {$\\cdots$};\n    \\draw[->,thick,headcolor] ([yshift=-2ex]a1.south)--(a1.south);\n    \\node[draw,circle,text=statecolor] (state) [below=2ex of a1.south] {$q_I$};\n  \\end{tikzpicture}\n  \\caption{The initial configuration of a Turing machine}\n  \\label{figure:tm:configuration}\n\\end{figure}\n\nTo explain how configurations are encoded as subtyping queries, let us first introduce some syntax (adopted from Grigore's paper).\nWe write a generic type \\inline{A<B<C>>} as $ABC$ for short.\nThe use of $\\fsubtype$ instead of $\\subtype$ in a subtyping query means that the type on the left-hand side should be read in reverse (the same goes for $\\fsuptype$ and $\\suptype$), e.g., $ABC \\fsubtype DE$ is equivalent to $CBA \\subtype DE$.\n\nThe initial TM configuration, depicted in \\cref{figure:tm:configuration}, is encoded by the following subtyping query:\n\\begin{equation}\\label{eq:initial:query}\n  {\\color{botcolor}ZEEL_ \\#}N{\\color{headcolor}M^L}N{\\color{symbcolor}L_{a_1}}N{\\color{symbcolor}L_{a_2}}N \\cdots\n  N{\\color{symbcolor}L_{a_m}}N{\\color{botcolor}L_ \\#}{\\color{statecolor}Q_I^{wR}}\n  \\fsubtype {\\color{botcolor}EEZ}\n\\end{equation}\nObserve that the types in \\cref{eq:initial:query} have the same colors as the machine configuration elements in \\cref{figure:tm:configuration} which they encode.\nFor example, the type $L_{a_1}$ encodes the tape symbol $a_1$, and both are colored in purple.\nAs the type on the left-hand side is written in reverse, it is possible to obtain the content of the encoded tape by reading the $L$ types from the left to the right.\nThe type $EEZ$ at both ends of the query encodes an infinite sequence of blank symbols.\nThe machine state $q_I$ is encoded by the type $Q^{wR}_I$, and the machine head by the type $M^L$ (the superscripts vary).\n\nGrigore referred to the subtyping query in \\cref{eq:initial:query} as a \\emph{subtyping machine} because when the subtyping algorithm tries to resolve it, it simulates the computation steps of the original TM.\nThe state type $Q_I$ is moved along the tape until it reaches the head type $M^L$.\nAt that point, the subtyping algorithm simulates a single TM transition by overwriting the current tape cell $L_{a_1}$, moving the machine head, and changing the machine state.\nThe resulting subtyping query correctly encodes the next configuration in the TM run.\nThis process continues until the machine accepts, and the query is resolved, or the machine aborts and a compilation error is raised.\nIf the machine runs indefinitely, the subtyping algorithm does not terminate.\n\nWhile TMs move the machine head to the left or right freely, subtyping machines can change direction only when reaching the end of the tape $EEZ$.\nAfter simulating a TM transition, the subtyping machine must reach the end(s) of the tape, rotate, and then reach the location of the machine head $M$ in the right orientation, before it can simulate the next transition.\nIn general, Grigore's subtyping machines can make $O(m)$ operations for every computation step of the TM they simulate, where $m$ is the number of symbols on the tape, resulting in a substantial slowdown.\nFor example, Grigore's simulation of the CYK algorithm, which usually runs in $O(n^3)$, takes $O(n^9)$ subtyping deduction steps to be completed.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the classical Boolean continuum \npercolation model (see \\cite{MR} for an overview), \none considers \na homogeneous Poisson process $\\eta$ of rate $\\lambda>0$ in ${\\mathbb{R}}^d$, \nand around each point $x\\in \\eta$ one places a ball $B(x,r)$ of radius $r.$ \nThe main object of study is then \n\\begin{equation} \\label{eqn:Cdef}\n{\\mathcal{C}}:=\\bigcup_{x\\in \\eta}B(x,r),\n\\end{equation}\nwhich is referred to as the {\\em occupied} set.\nIt is well known (see \\cite{MR}, Chapter 3) that there exists an \n$r_c=r_c(d)\\in(0,\\infty)$ such that \n\\[\nr_c:=\\inf\\{r:{\\mathbb{P}}({\\mathcal{C}}(r) \\textrm{ contains an unbounded component})>0\\}.\n\\]\nIt is also well known that \n${\\mathbb{P}}({\\mathcal{C}}(r) \\textrm{ contains an unbounded component})\\in \\{0,1\\}.$\nAn immediate scaling argument shows that varying $\\lambda$ is equivalent\nto varying $r,$ and so one can fix $\\lambda=1.$\nThis model was introduce by Gilbert in \\cite{Gilbert} and further studied \nin \\cite{Alexander}, \\cite{BenSchramm}, \\cite{MenshSid} and \\cite{Roy}\n(to name a few), while a dynamical version of this model was studied \nin \\cite{ABGM}.\n\nWe consider a natural extension of this model. \nLet $\\eta$ be a Poisson process with rate $\\lambda$ in\n${\\mathbb{R}}^d,$ and let $x\\in \\eta$ denote a point in this process\n(here we use the standard abuse of notation by writing \n$x\\in \\eta$ instead of $x\\in {\\rm supp}(\\eta)$).\nFurthermore, let $l:(0,\\infty) \\to (0,\\infty)$ be a non-increasing function \nthat we will call the {\\em attenuation} function.\nWe then define the random field \n$\\Psi=\\Psi(l,\\eta)$ at any point $y\\in {\\mathbb{R}}^d$ by\n\\begin{equation} \\label{eqn:Psidef}\n\\Psi(y):=\\sum_{x\\in \\eta}l(|x-y|).\n\\end{equation}\nIn order for this to be well defined at every point, we let \n$l(0):=\\lim_{r \\to 0}l(r)$ (which can possibly be infinite). \nOne can think of $\\Psi$ as a random potential field where the \ncontribution form a single point $x\\in \\eta$ is determined \nby the function $l.$\n\nFor any $0<h<\\infty,$ we define \n\\[\n\\Psi_{\\geq h}:=\\{y\\in {\\mathbb{R}}^d: \\Psi(y)\\geq h\\},\n\\]\nwhich is simply the part of the random field $\\Psi$  \nwhich is at or above the level $h.$ Sometimes we also need \n\\[\n\\Psi_{> h}:=\\{y\\in {\\mathbb{R}}^d: \\Psi(y)> h\\}.\n\\]\n\nWe note that if we consider our general model with $l(|x|)=I(|x|\\leq r)$ \n(where $I$ is an indicator function), we have that \n${\\mathcal{C}}$ and $\\Psi_{\\geq 1}$ have the same distribution, so the Boolean \npercolation model can be regarded \nas a special case of our more general model. \n\nWhen $l$ has unbounded support, adding or removing\na single point of $\\eta$ will affect the field $\\Psi$ at every point\nof ${\\mathbb{R}}^d.$ Thus, our model does not have a so-called finite\nenergy condition\nwhich is the key \nto many standard proofs\nin percolation theory. This is what makes studying $\\Psi$\nchallenging (and in our opinion interesting). However, if we assume \nthat $l$ has bounded support, a version of finite energy is recovered \n(see also the remark in Section \\ref{sec:uniqueness} after the proof of \nour uniqueness result, Theorem \\ref{thm:uniqueness}).\n\n\nIt is easy to see that varying $h$\nand varying $\\lambda$ is {\\em not} equivalent. However, we will \nnevertheless restrict our attention to the case $\\lambda=1$. \nIn fact, there are many different sub-cases \nand generalizations that can be studied.\nFor instance: We can let $\\lambda\\in {\\mathbb{R}},$ we can study $l$ having bounded\nor unbounded support, we can let $l$ be a bounded or unbounded function,\nlet $l$ be continuous or discontinuous and we can study $\\Psi_{\\geq h}$\nor $\\Psi_{> h}$, to name a few possibilities.\nWhile some results (Theorems \\ref{thm:hcfinite} and \\ref{thm:hcNT}) \ninclude all or most of the cases listed, others \n(Proposition \\ref{prop:contfield} and Theorems \\ref{thm:uniqueness}\nand Theorem \\ref{thm:thetacont}) require more specialized proofs.\nThe purpose of this paper is {\\em not} to handle all different cases. \nInstead, we will focus on the extension of the classical Boolean percolation model\nthat we find to be the most natural\nand interesting; when $l$ is continuous and with \nunbounded support.\\\\\n\n\nWe will now proceed to state our results, but first we \nhave the following natural definition.\n\\begin{definition}\nIf $\\Psi_{\\geq h}$ ($\\Psi_{> h}$) contains an unbounded connected component, \nwe say that \n$\\Psi$ {\\em percolates} at (above) level $h,$ or simply that $\\Psi_{\\geq h}$\n($\\Psi_{> h}$) percolates.\n\\end{definition}\nOne would of course expect that percolation occurs either with probability 0 \nor with probability 1, and indeed, our next result shows just that.\n\\begin{proposition} \\label{prop:perc01}\nWe have that \n\\[\n{\\mathbb{P}}(\\Psi_{\\geq h} \\textrm{ percolates})\\in\\{0,1\\}\n\\textrm{ and } {\\mathbb{P}}(\\Psi_{>h} \\textrm{ percolates})\\in\\{0,1\\}.\n\\]\n\\end{proposition}\n\n\\medskip\\noindent\n{\\bf Proof.} \nThis follows from a classical \nergodicity argument. Indeed, the random field $\\Psi$ is ergodic with \nrespect to the \ngroup of translations of the space, see for instance the argument \nin \\cite{MR}, Section 2.1, \nwhere it is formulated for the Boolean model and the random connection \nmodel, but the \nargument applies to our case as well. Since the event that \n$\\Psi_{\\geq h}$ percolates \nis invariant under translations, it must then have probability 0 or 1. \n\\fbox{}\\\\\n\nWe now define\n\\[\nh_c:=\\sup\\{h:\\Psi_{\\geq h} \\textrm{ percolates with probability 1}\\}.\n\\]\nIf we define $\\tilde{h}_c$ as above, but with $\\Psi_{\\geq h}$\nreplaced by $\\Psi_{>h},$ we see that $h_c=\\tilde{h}_c.$ Indeed, \nif $h<h_c$ then $h\\leq \\tilde{h}_c$ so that $h_c\\leq \\tilde{h}_c.$\nAlso, if $h_c<\\tilde{h}_c$ then we can find $h_c<g<\\tilde{h}_c$\nso that $\\Psi_{>g}$ percolates while $\\Psi_{\\geq g}$ does not.\nThis is clearly impossible since $\\Psi_{\\geq g} \\subset \\Psi_{>g}.$\n\nOne of the main efforts of this paper is to establish conditions \nunder which $h_c$ is nontrivial. \nAs we will see, our results are qualitatively different depending on \nwhether the attenuation function $l$ has bounded support or not. \nOur first main result is the following. \n\n\\begin{theorem} \\label{thm:hcfinite}\nIf the attenuation function $l$ satisfies $\\int_{1}^\\infty r^{d-1}l(r)dr<\\infty,$\nthen $h_c<\\infty.$ If instead $\\int_{1}^\\infty r^{d-1}l(r)dr=\\infty,$\nthen almost surely $\\Psi(y)=\\infty$ for every $y\\in {\\mathbb{R}}^d$, and so $h_c=\\infty.$\n\\end{theorem}\n\\noindent\n{\\bf Remark:} The choice of the lower integral boundary 1 in $\\int_{1}^\\infty r^{d-1}l(r)dr$ \nis somewhat arbitrary, as replacing it with $\\int_{c}^\\infty r^{d-1}l(r)dr$ for any \n$0<c<\\infty,$ would give the same result. \nNote also that if $l$ has bounded support, we must have that \n$\\int_{1}^\\infty r^{d-1}l(r)dr<\\infty.$\n\\bigskip\n\nOur next result concerns the positivity of $h_c.$ The proof is \nvery straightforward and complements Theorem \\ref{thm:hcfinite}. \nWe define $r_l:=\\sup\\{r:l(r)>0\\}$.\n\n\\begin{theorem} \\label{thm:hcNT}\nFor $d=2,$ then $h_c>0$ iff $r_l> r_c.$\nFor $d\\geq 3,$ $h_c>0$ if $r_l> r_c$ while\n$h_c=0$ if $r_l< r_c.$\n\\end{theorem}\n\\noindent\n{\\bf Remark:} As is clear from the proof, the gap when $r_l=r_c$\nfor $d\\geq 3,$ is simply due to the fact that for $d \\geq 3$ it is unknown whether \n${\\mathbb{P}}({\\mathcal{C}}(r_c) \\textrm{ contains an unbounded component})$ is 0 (as when $d=2)$ \nor 1.\n \n\\bigskip\n\nWe highlight our interest in the case when $l$ has unbounded support by formulating the following \nimmediate corollary.\n\\begin{corollary}\nIf the attenuation function $l$ has unbounded support, \nthen $0<h_c<\\infty$ if and only if \n $\\int_{1}^\\infty r^{d-1}l(r)dr<\\infty.$\n\\end{corollary}\nFor the rest of this section, we will only consider functions $l$\nsatisfying $\\int_1^\\infty r^{d-1}l(r)dr<\\infty$.\n\nWe remark that mere a.s.\\ finiteness of the field $\\Psi$ does not guarantee that \nthere is a nontrivial phase transition. Indeed, one can construct an example \n(following the examples of \\cite{MR} Chapter 7.7)\nof a stationary process together with a suitable attenuation function so that the \nensuing field is finite a.s., while $h_c=\\infty.$ \n\nNext, we turn to the everywhere continuity of the field $\\Psi$. \nOf course, if $l$ is not continuous, everywhere continuity of $\\Psi$ cannot hold. \nHowever, if $l$ is continuous then everywhere \ncontinuity of the field $\\Psi$ would be expected. We note that in the case of \n$l$ being unbounded (i.e. $\\lim_{r \\to 0}l(r)=\\infty$), we simply define\n$\\Psi(y)=\\infty$ for every $y\\in \\eta.$\n\n\\begin{proposition} \\label{prop:contfield}\nIf $l$ is continuous, then\nthe random field $\\Psi$ is a.s.\\ everywhere continuous.\n\\end{proposition}\n\nProposition \\ref{prop:contfield}\nwill be of use when proving our next result concerning uniqueness of the \nunbounded component.\n\n\\begin{theorem}\n\\label{thm:uniqueness}\nLet $h$ be such that $\\Psi_{\\geq h}$ ($\\Psi_{>h}$) contains an \nunbounded component.\nIf $l$ is continuous and with unbounded support, then for $d=2$,\nthere is a unique such unbounded component.\n\\end{theorem}\n\\noindent\n{\\bf Remarks:} \nWe will prove this theorem first for $\\Psi_{>h}$ and then infer it for \n$\\Psi_{\\geq h}$, see also the discussion before the proof of the theorem.\n\nThere are of course a number of possible generalizations of \nthis statement, and perhaps the most interesting\/natural would be to investigate\nit for $d\\geq 3.$ We discuss this in some detail after the proof of \nTheorem \\ref{thm:uniqueness}. \\\\\n\nLet ${\\mathcal{C}}_{o,\\geq}(h)$ (${\\mathcal{C}}_{o,>}(h)$) be the connected component of \n$\\Psi_{\\geq h}$ ($\\Psi_{> h}$) that contains the origin $o.$ Define \nthe percolation function\n\\[\n\\theta_{\\geq}(h):={\\mathbb{P}}({\\mathcal{C}}_{o,\\geq}(h) \\textrm{ is unbounded}),\n\\]\nand similarly define $\\theta_>(h)$. Our last result is the following.\n\n\\begin{theorem} \\label{thm:thetacont}\nThe functions $\\theta_{\\geq}(h)$ and $\\theta_{>}(h)$ are equal and \ncontinuous for $h<h_c$.\n\\end{theorem}\n\n\n\\medskip\n\nThe rest of the paper is organized as follows. \nIn Section \\ref{sec:phasetransition}\nwe prove Theorems \\ref{thm:hcfinite} and \\ref{thm:hcNT}. The continuity of \n$\\Psi$ (Proposition \\ref{prop:contfield})\nis proved in Section \\ref{sec:cont} and this results is then used in \nSection \\ref{sec:uniqueness} in order to prove Theorem \\ref{thm:uniqueness},\nwhich in turn will allow us to prove Theorem \\ref{thm:thetacont}.\n\n\n\\section{Conditions for the non-triviality of $h_c$} \\label{sec:phasetransition}\nWe start by proving Theorem \\ref{thm:hcNT} as this is easily done. \n\n\\medskip\n{\\bf Proof of Theorem \\ref{thm:hcNT}.}\nRecall the constructions \\eqref{eqn:Cdef} and \\eqref{eqn:Psidef}, and observe \nthat by using the same realization of $\\eta,$ we can couple $\\Psi$ and ${\\mathcal{C}}$\non the same probability space in the natural way. By using this coupling, we see that \nfor any $h>0,$ $\\Psi_{\\geq h}\\subset {\\mathcal{C}}(r_l).$ In the case $d=2,$\nit is known (see \\cite{MR}, Theorem 4.5) that ${\\mathcal{C}}(r_c)$ does not \npercolate, showing that $h_c=0$ when $d=2$ and $r_l\\leq r_c.$\nFor $d\\geq 3,$ the statement follows by \nobserving that ${\\mathcal{C}}(r)$ does not percolate for $r<r_c$ by definition \nof $r_c.$\n\nAssume instead that $r_l> r_c.$ Let $r_c<r<r_l,$ and  $h=h(r)$\nbe any $h$ such that \n\\[\nB(0,r) \\subset \\{y:l(|y|)\\geq h\\}.\n\\]\nWith this choice of $h,$ we see that any point $y$ within radius $r$ from some \n$x\\in \\eta$ will also belong to $\\Psi_{\\geq h}$ so that\n\\[\n{\\mathcal{C}}(r)\\subset \\Psi_{\\geq h}.\n\\]\nSince $r>r_c,$ ${\\mathcal{C}}(r)$ a.s.\\ contains an unbounded component and hence so does \n$\\Psi_{\\geq h}.$\n\\fbox{}\\\\\n\nThe proof of Theorem \\ref{thm:hcfinite} is much more involved, and will require\na number of preliminary lemmas to be established first. In order to see what the \npurpose of these will be, we start by giving an outline of the \nstrategy of our proof along with introducing some of the relevant notation.\nLet $\\alpha {\\mathbb{Z}}^d$ denote the lattice with \nspacing $\\alpha>0.$ For any $z\\in \\alpha {\\mathbb{Z}}^d,$ let $B(z,\\alpha)$\ndenote the closed box of side length $\\alpha$ centered at $z,$ and define\n${\\mathcal{B}}_\\alpha:=\\{B(z,\\alpha):z\\in \\alpha {\\mathbb{Z}}^d\\}.$\nFor convenience, we assume from now on that $\\alpha<1.$ \n \n\\medskip\\noindent\n{\\bf Claim:} There exists an $\\epsilon>0$ such that if for \nany $0<\\alpha<1$ and every $k$ and \ncollection of distinct cubes $B_1,\\ldots,B_k\\in {\\mathcal{B}}_\\alpha,$ we have that\n\\begin{equation} \\label{eqn:epsest}\n{\\mathbb{P}}(\\sup_{y\\in B_1}\\Psi(y)\\geq h, \\ldots, \\sup_{y\\in B_k}\\Psi(y)\\geq h)\n\\leq \\epsilon^k,\n\\end{equation}\nthen $\\Psi_{\\geq h}$ does a.s.\\ not contain an unbounded component. \\\\\n\nThis claim can be proved using standard percolation arguments as follows. \nLet $B^o\\in {\\mathcal{B}}_\\alpha$ be the cube containing the origin $o$ and let \n${\\mathcal{O}}$ denote the event that $B^o$ intersects an\nunbounded component of $\\Psi_{\\geq h}$. If ${\\mathcal{O}}$ occurs, then\nfor any $k,$ there must exist a sequence\n$B_1,B_2,\\ldots B_k\\in {\\mathcal{B}}_\\alpha$ such that $B_1=B^o,$\n$B_i \\neq B_j$ for every $i\\neq j,$  $B_i\\cap B_{i+1}\\neq \\emptyset$\nfor every $i=1,\\ldots,k-1$ and with the property that \n$\\sup_{y\\in B_i}\\Psi(y)\\geq h$ for every $i=1,\\ldots,k.$\nWe note that the number of such paths must be bounded\nby $3^{dk}$, as any box has fewer than $3^d$ 'neighbors'. \nThus, from \\eqref{eqn:epsest} we get that \n${\\mathbb{P}}({\\mathcal{O}})\\leq 3^{dk}\\epsilon^k,$ and since this holds for\narbitrary $k$ this proves the claim by taking $\\epsilon<3^{-d}$.\n\nOne issue when proving \\eqref{eqn:epsest} is that we want\nto consider the supremum of the field within the boxes $B_1,\\ldots,B_k.$\nHowever, this is fairly easily dealt with by introducing an auxiliary \nfield $\\tilde{\\Psi}$ with the property that for any $B\\in {\\mathcal{B}}_\\alpha$\n$\\tilde{\\Psi}(y_c(B))\\geq \\sup_{y\\in B} \\Psi(y)$ where $y_c(B)$\ndenotes the center of $B$ (see further \\eqref{eqn:Psisup}). This allows us\nto consider $k$ fixed points of the new field $\\tilde{\\Psi}$ rather than \nthe supremums involved in \\eqref{eqn:epsest}.\n\nOne of the main problems in proving \\eqref{eqn:epsest} is the \nlong range dependencies involved whenever $l$ has unbounded support \n(as discussed in the introduction). The strategy to resolve this issue\nis based on the simple observation that \n\\begin{equation} \\label{eqn:supsumbound}\n\\left\\{\\sup_{y\\in B_1}\\Psi(y)\\geq h, \\ldots, \\sup_{y\\in B_k}\\Psi(y)\\geq h\\right\\}\n\\subset \\left\\{ \\sum_{l=1}^k \\tilde{\\Psi}(y_c(B_l))\\geq kh \\right\\}.\n\\end{equation}\nThe event on the right hand side of \\eqref{eqn:supsumbound} can be analyzed \nusing a version of Campbell's theorem (see e.g.\\ \\cite{Kingman} p. 57-57). An obvious\nproblem with this is that if $l$ is unbounded and if a single point of \n$\\eta$ falls in $\\bigcup_{l=1}^k B_l$, then the sum in \\eqref{eqn:supsumbound} is \ninfinite. However, by letting $\\alpha$ above be very small, we can make sure that \nwith very high probability, ``most'' of the boxes $B_1,\\ldots,B_k$ will \nnot contain any points of $\\eta$ (and in fact there will not even be a point\nin a certain neighborhood of the box). \nWe then use a more sophisticated version of \n\\eqref{eqn:supsumbound} (i.e. \\eqref{eqn:prelest}) where we condition \non which of the boxes $B_1,\\ldots,B_k$ have a point of $\\eta$ in their neighborhood,\nand then sum only over the boxes whose neighborhoods are vacant of points.\nThis in turn introduces another problem, namely that we now have to deal \nwith a Poisson process conditioned on the presence and absence of points\nof $\\eta$ in the neighborhoods of the boxes $B_1,\\ldots,B_k.$ \nIn particular, we have to control the\ndamage from knowing the presence of such points. This is the purpose of \nLemmas \\ref{lemma:elemPoisdom} and \\ref{lemma:elemPoisdom2}, which will tell\nus that our knowledge is not worse than having no information at all plus\nadding a few extra points to the process. Later, Lemma \n\\ref{lemma:bndedfield} will enable us to control the effect of this addition \nof extra points.\n\n\\bigskip\n\nWe now start presenting the rigorous proofs. Our first lemma \nis elementary, and the result is presumably folklore. However,\nwe give a proof for sake of completeness.\n\\begin{lemma} \\label{lemma:elemPoisdom}\nLet $X$ be a Poisson distributed random variable with parameter $\\lambda.$ \nWe have that for any $k\\geq 0,$\n\\begin{equation}\\label{eqn:Poiconddom}\n{\\mathbb{P}}(X\\geq k | X\\geq 1)\\leq {\\mathbb{P}}(X\\geq k-1).\n\\end{equation}\n\\end{lemma}\n\n\\noindent\n{\\bf Proof.}\nWe claim that for any $X^n\\sim$ Bin($n,p$) where $np=\\lambda,$ and \nany $0\\leq k \\leq n,$ we have that \n\\begin{equation} \\label{eqn:Binconddom}\n{\\mathbb{P}}(X^n\\geq k| X^n \\geq 1)\\leq {\\mathbb{P}}(X^n\\geq k-1).\n\\end{equation}\nWe observe that from \\eqref{eqn:Binconddom} we get that \n\\[\n{\\mathbb{P}}(X\\geq k | X\\geq 1)\n=\\lim_{n \\to \\infty}{\\mathbb{P}}(X^n\\geq k| X^n \\geq 1)\n\\leq \\lim_{n \\to \\infty}{\\mathbb{P}}(X^n\\geq k-1)= {\\mathbb{P}}(X \\geq k-1).\n\\]\nThis establishes \\eqref{eqn:Poiconddom}, and so we need to prove \n\\eqref{eqn:Binconddom}.\n\nWe will prove \\eqref{eqn:Binconddom} through induction, and we start by observing\nthat it trivially holds for $n=1$ and $k=0,1.$ Assume therefore that \n\\eqref{eqn:Binconddom} holds for $n$ and any $k=0,\\ldots,n.$\nWe will write $X^n=X_1+\\ldots+X_{n}$ where $\\left(X_i\\right)_{i\\geq 1}$ is an \ni.i.d. sequence with ${\\mathbb{P}}(X_i=1)=1-{\\mathbb{P}}(X_i=0)=p.$ Of course, \n\\eqref{eqn:Binconddom} trivially holds for $n+1$ and $k=0.$ Furthermore, \nwe have that for any $k=1,\\ldots,n,$\n\\begin{eqnarray*}\n\\lefteqn{{\\mathbb{P}}(X^{n+1} \\geq k | X^{n+1}\\geq 1)}\\\\\n& & ={\\mathbb{P}}(X^{n+1} \\geq k | X^{n+1}\\geq 1, X_{n+1}=1){\\mathbb{P}}(X_{n+1}=1 |X^{n+1}\\geq 1)\\\\\n& & \\ \\ \\ \\ +{\\mathbb{P}}(X^{n+1} \\geq k | X^{n+1}\\geq 1, X_{n+1}=0){\\mathbb{P}}(X_{n+1}=0 |X^{n+1}\\geq 1)\\\\\n& & ={\\mathbb{P}}(X^{n} \\geq k-1){\\mathbb{P}}(X_{n+1}=1 |X^{n+1}\\geq 1)\n +{\\mathbb{P}}(X^{n} \\geq k | X^{n}\\geq 1){\\mathbb{P}}(X_{n+1}=0 |X^{n+1}\\geq 1)\\\\\n& & \\leq {\\mathbb{P}}(X^{n} \\geq k-1),\n\\end{eqnarray*}\nwhere we use the induction hypothesis that ${\\mathbb{P}}(X^{n} \\geq k | X^{n}\\geq 1)\n\\leq {\\mathbb{P}}(X^{n} \\geq k-1)$ in the last inequality. Finally, \n\\[\n{\\mathbb{P}}(X^{n+1} =n+1 | X^{n+1}\\geq 1)\n={\\mathbb{P}}(X^{n} = n){\\mathbb{P}}(X_{n+1}=1 |X^{n+1}\\geq 1)\n\\leq {\\mathbb{P}}(X^{n} = n),\n\\]\nand this establishes \\eqref{eqn:Binconddom} for $n+1$ and any $k=0,\\ldots,n+1.$\n\\fbox{}\\\\\n\nLet $A_1,A_2,\\ldots,A_n$ be subsets of ${\\mathbb{R}}^d, $ and let $C_1,\\ldots,C_m$\nbe a partition of $\\cup_{i=1}^n A_i$ such that for any $i,$ \n\\begin{equation} \\label{eqn:Aipart}\nA_i=\\cup_{k=1}^l C_{i_k},\n\\end{equation}\nfor some collection $C_{i_1},\\ldots,C_{i_l}.$ Let $\\eta_A$ be a homogeneous \nPoisson process of rate $\\lambda>0$ on $\\cup_{i=1}^n A_i$ conditioned \non the event $\\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\},$ and \nlet $\\eta'_A$ be a homogeneous (unconditioned) Poisson process of \nrate $\\lambda>0$ on $\\cup_{i=1}^n A_i.$ Furthermore, let \n$\\xi_A$ be a point process on $\\cup_{i=1}^n A_i$ consisting of\nexactly one point in each of the sets $C_1,\\ldots,C_m$ such that \nthe position of the point in $C_i$ is uniformly distributed within \nthe set, and so that this position is independent between sets.\n\nOur next step is to use Lemma \\ref{lemma:elemPoisdom} to prove \na result relating the conditioned Poisson process $\\eta_A$ to \nthe sum $\\eta'_A+\\xi_A,$ where $\\eta'_A$ and $\\xi_A$ are independent.\nFor two point processes $\\eta_1,\\eta_2$ in ${\\mathbb{R}}^d$, we write $\\eta_1\\preceq \\eta_2$ if there exists a coupling \nof $\\eta_1,\\eta_2$ so that ${\\mathbb{P}}(\\eta_1\\subset \\eta_2)=1.$\n\\begin{lemma} \\label{lemma:elemPoisdom2}\nLet $\\eta_A,\\eta'_A$ and $\\xi_A$ be as above, and let \n$\\eta'_A$ and $\\xi_A$ be independent. We have that \n\\[\n\\eta_A \\preceq \\eta'_A+\\xi_A.\n\\]\n\\end{lemma}\n\nInformally, Lemma \\ref{lemma:elemPoisdom2} tells us that if \nwe consider a homogeneous Poisson process conditioned on the presence\nof points in $A_1,\\ldots,A_k$, it is not worse than taking an unconditioned\nprocess and adding single points to all the sets $C_1,\\ldots,C_m$ (which are \nused as the building blocks for the sets $A_1,\\ldots,A_k$).\n\n\\medskip\n\n\\noindent\n{\\bf Proof of Lemma \\ref{lemma:elemPoisdom2}.}\nAs usual, let $\\eta$ be a homogeneous Poisson process on ${\\mathbb{R}}^d.$\nLet $J=(j_1,\\ldots,j_m)\\in\\{0,1\\}^m$ and define\n\\[\n{\\mathcal{C}}_J=\\bigcap_{l:j_l=1}\\{\\eta(C_l)\\geq 1\\}\\bigcap_{l:j_l=0}\\{\\eta(C_l)=0\\}.\n\\]\nWe note that either ${\\mathcal{C}}_J \\subset \\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\},$ or \n${\\mathcal{C}}_J \\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\}=\\emptyset,$ which follows from \n\\eqref{eqn:Aipart}. Indeed, if for any $i$ and $J\\in \\{0,1\\}^m,$ \nall of the sets $C_{i_k}$ in \n\\eqref{eqn:Aipart} have $\\eta(C_{i_k})=0$, then \n${\\mathcal{C}}_J \\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\}=\\emptyset.$ On the other hand, if for \nevery $i,$ there exists some set $C_{i_k}$ in \\eqref{eqn:Aipart} such that \n$\\eta(C_{i_k})=1$ this implies that $\\{\\eta(A_i)\\geq 1\\}$ \noccurs for every $i.$\n\nUsing this, we have that for any $(k_1,\\ldots,k_m)\\in {\\mathbb{N}}^m,$ \n\\begin{eqnarray} \\label{eqn:CJineq}\n\\lefteqn{{\\mathbb{P}}(\\eta(C_1)\\geq k_1,\\ldots, \\eta(C_m)\\geq k_m \n|\\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\})} \\\\\n& & =\\sum_{J\\in \\{0,1\\}^m}{\\mathbb{P}}(\\eta(C_1)\\geq k_1,\\ldots, \\eta(C_m)\\geq k_m \n|{\\mathcal{C}}_J){\\mathbb{P}}({\\mathcal{C}}_J | \\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\}), \\nonumber\n\\end{eqnarray}\nsince ${\\mathbb{P}}({\\mathcal{C}}_J | \\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\})=0$ if \n${\\mathcal{C}}_J \\not \\subset \\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\}.$\nFurthermore, for any $J\\in \\{0,1\\}^m$\n\\begin{equation} \\label{eqn:CJineq2}\n{\\mathbb{P}}(\\eta(C_1)\\geq k_1,\\ldots, \\eta(C_m)\\geq k_m |{\\mathcal{C}}_J)\n\\leq {\\mathbb{P}}(\\eta(C_1)\\geq k_1-1,\\ldots, \\eta(C_m)\\geq k_m-1),\n\\end{equation}\nby using Lemma \\ref{lemma:elemPoisdom} and a trivial bound.\n\nCombining \\eqref{eqn:CJineq} and \\eqref{eqn:CJineq2} yields\n\\begin{eqnarray*}\n\\lefteqn{{\\mathbb{P}}(\\eta_A(C_1)\\geq k_1,\\ldots, \\eta_A(C_m)\\geq k_m )}\\\\\n& & ={\\mathbb{P}}(\\eta(C_1)\\geq k_1,\\ldots, \\eta(C_m)\\geq k_m \n|\\cap_{i=1}^n\\{\\eta(A_i)\\geq 1\\})\\\\\n& & \\leq \n{\\mathbb{P}}(\\eta(C_1)\\geq k_1-1,\\ldots, \\eta(C_m)\\geq k_m-1)\\\\\n& & ={\\mathbb{P}}((\\eta'_A+\\xi_A)(C_1)\\geq k_1,\\ldots, (\\eta'_A+\\xi_A)(C_m)\\geq k_m).\n\\end{eqnarray*}\nThe statement follows by the elementary property of a Poisson process,\nthat conditioned on a certain number of points falling within a fix set\nD, these points are independently and uniformly distributed within that set.\n\\fbox{}\\\\\n\n\nWe now turn to the issue of taking the\nsupremum of the field over a box. Therefore, let $0<\\alpha<1,$ and\ndefine the auxiliary attenuation function $\\tilde{l}_\\alpha$ by \n\\[\n\\tilde{l}_\\alpha(r)=\n\\left\\{\n\\begin{array}{ll}\nl(0) & \\textrm{if } r\\leq \\alpha \\sqrt{d}\/2 \\\\\nl(r-\\alpha\\sqrt{d}\/2) & \\textrm{if } r\\geq \\alpha\\sqrt{d}\/2,\n\\end{array}\n\\right.\n\\]\nfor every $r\\geq 0.$ If $y_{c}(B)$ denotes the center of the box $B\\in {\\mathcal{B}}_\\alpha,$\nwe note that for any $y\\in B$ and $x\\in {\\mathbb{R}}^d,$ \n\\[\n\\tilde{l}_\\alpha(|x-y_c(B)|)\n\\geq \\tilde{l}_\\alpha(|x-y|+|y-y_c(B)|)\n\\geq \\tilde{l}_\\alpha(|x-y|+\\alpha\\sqrt{d}\/2)\n= l(|x-y|).\n\\]\nTherefore, if we let $\\tilde{\\Psi}$ be the field we get by using $\\tilde{l}$\nin place of $l$ in \\eqref{eqn:Psidef}, we get that \n\\begin{equation}\\label{eqn:Psisup}\n\\tilde{\\Psi}(y_c(B))=\\sum_{x\\in \\eta}\\tilde{l}_\\alpha(|x-y_c(B)|)\n\\geq \\sup_{y\\in B}\\sum_{x\\in \\eta} l(|x-y|)=\\sup_{y\\in B} \\Psi(y).\n\\end{equation}\n\nOur next lemma will be a central ingredient of the proof of \nTheorem \\ref{thm:hcfinite}. It will deal with the effect to the field \n$\\tilde{\\Psi}$ of adding extra points to $\\eta$. To that end, \nlet $A_o$ be the box of side length $\\alpha(4\\lceil\\sqrt{d}\\rceil+1)$ \ncentered around the origin $o$. For any box $B\\in {\\mathcal{B}}_\\alpha$ with \n$B\\cap A_o=\\emptyset,$\nplace a point $x_B$ in $B$ at the closest distance to the origin, and let \n$\\xi$ denote the corresponding (deterministic) point set. Let\n\\[\n\\tilde{\\Psi}_{A_o}(y):=\\sum_{x \\in \\xi}\\tilde{l}_\\alpha(|x-y|),\n\\]\nbe the corresponding deterministic field. \n\n\\begin{lemma} \\label{lemma:bndedfield}\nThere exists a constant $C<\\infty$ depending on $d$ but not on $\\alpha$ and\nsuch that for every $0<\\alpha<1,$\n\\[\n\\tilde{\\Psi}_{A_o}(o)\\leq \\frac{C}{\\alpha^d} I_\\alpha,\n\\]\nwhere \n\\[\nI_\\alpha=\\int_{\\alpha\/2}^\\infty r^{d-1} l(r)dr<\\infty.\n\\]\n\\end{lemma}\n\\noindent\n{\\bf Proof.}\nConsider some $B\\in {\\mathcal{B}}_\\alpha$ such that $B\\cap A_o=\\emptyset.$\nWe have that \n\\[\n\\tilde{l}_\\alpha(|x_B|)\\leq \\frac{1}{Vol(B)}\\int_{B}\\tilde{l}_\\alpha(|x|-{\\rm diam}(B))dx\n=\\frac{1}{\\alpha^d}\\int_{B}\\tilde{l}_\\alpha(|x|-\\alpha \\sqrt{d})dx.\n\\]\nTherefore, \n\\begin{eqnarray*}\n\\lefteqn{\\tilde{\\Psi}_{A_o}(o)\n\\leq \\frac{1}{\\alpha^d}\\int_{{\\mathbb{R}}^d\\setminus A_o}\\tilde{l}_\\alpha(|x|-\\alpha \\sqrt{d})dx}\\\\\n& & \\leq \\frac{C}{\\alpha^d}\\int_{\\alpha(2\\lceil\\sqrt{d}\\rceil+1\/2)}^\\infty\nr^{d-1}l(r-2\\alpha \\sqrt{d})dr \\\\\n& & \\leq \\frac{C}{\\alpha^d}\\int_{\\alpha\/2}^\\infty (r+\\alpha(2\\lceil\\sqrt{d}\\rceil))^{d-1}\nl(x) dr \\nonumber \\\\\n& & \\leq \\frac{C}{\\alpha^d}\\int_{\\alpha\/2}^\\infty (r+r(4\\sqrt{d}+2))^{d-1}\nl(r) dr \\nonumber \\\\\n& & = \\frac{C}{\\alpha^d}\\int_{\\alpha\/2}^\\infty r^{d-1} l(r)dr\n\\end{eqnarray*}\nwhere the constant $C=C(d)<\\infty$\nis allowed to vary in the steps of the calculations. \nFinally, the fact that $I_\\alpha<\\infty,$ follows easily from the fact that \n$\\int_{1}^\\infty r^{d-1} l(r)dr<\\infty.$\n\\fbox{}\\\\\n\n\nWe have now established all necessary tools in order to prove Theorem \n\\ref{thm:hcfinite}. However, since the proofs of the two statements of \nTheorem \\ref{thm:hcfinite} are very different, we start by proving the \nfirst one as a separate result.\n\n\\begin{theorem}\\label{thm:hcfinite_aux}\nIf $\\int_1^\\infty l^{d-1}l(r)<\\infty,$ then $h_c<\\infty.$\n\\end{theorem}\n\\noindent\n{\\bf Proof.}\nWe shall prove that for any $\\epsilon>0,$ \\eqref{eqn:epsest} holds\nfor $\\alpha$ small enough and $h$ large enough. This will prove our\nresult as explained just below \\eqref{eqn:epsest}.\n\nFor any $B\\in {\\mathcal{B}}_\\alpha,$ let $A_\\alpha(B)$ be the box concentric \nto $B$ and with side length $\\alpha(4\\lceil \\sqrt{d}\\rceil+1)$.\nLet $E(B)$ be the event that $\\eta(A_\\alpha(B))=0,$ and observe that \nif $c={\\mathbb{P}}(E(B)),$\nwe have that $c=c(\\alpha)\\to 1$ as $\\alpha \\to 0$. We say that the box \n$B$ is {\\em good} if the event $E(B)$ occurs. Goodness of the boxes \n$B\\in {\\mathcal{B}}_\\alpha$ naturally induces a percolation model on ${\\mathbb{Z}}^d$ with \na finite range dependency. Since the marginal probability $c(\\alpha)$ of being \ngood can be made to be arbitrarily close to 1 by taking \n$\\alpha$ small enough, we can use Theorem B26 of \\cite{SIS}\nto dominate an i.i.d.\\ product measure with density $p=p(\\alpha)$ on \nthe boxes $B\\in {\\mathcal{B}}_\\alpha.$\nFurthermore, by the same theorem, we can take $p(\\alpha)\\to 1$ as $\\alpha \\to 0.$\n\nFix $k$ and a collection $B_1,B_2,\\ldots, B_k$ as in \\eqref{eqn:epsest}. \nFor any $B_i,$ let $A_i=A_\\alpha(B_i)$, and let $\\Gamma_i:=I(B_i)$ \nwhere $I$ denotes an indicator function.\nIf we take $\\Gamma=\\sum_{i=1}^k \\Gamma_i,$ then\nby the above domination of a product measure of density $p,$ \nwe see that $\\Gamma$ is \nstochastically larger than $\\Gamma'\\sim$Bin$(p,k).$ Furthermore, \nwe have that \n\\begin{eqnarray*}\n\\lefteqn{{\\mathbb{P}}\\left(\\Gamma'\\leq \\frac{k}{2}\\right)}\\\\\n& & ={\\mathbb{P}}\\left(e^{\\log(1-p)\\Gamma'}\\geq e^{\\log(1-p)k\/2}\\right)\n\\leq e^{-\\log(1-p)k\/2}{\\mathbb{E}}\\left[e^{\\log(1-p)\\Gamma'}\\right]\\\\\n& & =e^{-\\log(1-p)k\/2}\\left(pe^{\\log(1-p)}+1-p\\right)^k\n\\leq 2^k e^{\\log(1-p)k\/2}=e^{-d(\\alpha)k},\n\\end{eqnarray*}\nwhere we can take $d(\\alpha)\\to \\infty$ as $\\alpha \\to 0,$ by \ntaking $p(\\alpha)\\to 1.$\nIf we define $G_k$ to be the event that at least $k\/2$ of the boxes \n$B_1,B_2,\\ldots, B_k$\nare good, we thus have that ${\\mathbb{P}}(G_k)\\geq 1-e^{-d(\\alpha)k}.$\n\nLet $J=J(\\eta)\\in\\{0,1\\}^k$ be such that $J_j=1$ iff $B_j$ is good, and identify \n$J$ with the corresponding subset of $\\{1,\\ldots,k\\}.$ Thus we write $j\\in J$\niff $B_j$ is good. For any fixed\n$J\\in \\{0,1\\}^k,$ we let $\\CD_J$ denote the event \n\\[\n\\bigcap_{j\\in J} E_j \\bigcap_{j\\in J^c} E_j^c\n\\]\nso that $\\CD_J$ is the event that each set $A_j$ such that $j\\in J$\nis vacant of points, while each set $A_j$ such that $j\\in J^c$\ncontains at least one point of $\\eta$.\nWe then have that \n\\begin{eqnarray} \\label{eqn:prelest}\n\\lefteqn{{\\mathbb{P}}(\\sup_{y\\in B_1}\\Psi(y)\\geq h, \\ldots, \\sup_{y\\in B_k}\\Psi(y)\\geq h)}\\\\\n& & \\leq \\sum_{J\\in \\{0,1\\}^k}\n{\\mathbb{P}}(\\tilde{\\Psi}(y_c(B_1))\\geq h,\\ldots,\\tilde{\\Psi}(y_c(B_k))\\geq h|\\CD_J){\\mathbb{P}}(\\CD_J)\\nonumber \\\\\n& & \\leq \\sum_{|J|\\geq k\/2}\n{\\mathbb{P}}(\\tilde{\\Psi}(y_c(B_1))\\geq h,\\ldots,\\tilde{\\Psi}(y_c(B_k))\\geq h|\\CD_J)\n{\\mathbb{P}}(\\CD_J)+e^{-d(\\alpha)k}, \\nonumber \n\\end{eqnarray}\nby using \\eqref{eqn:Psisup} in the first inequality and that \n${\\mathbb{P}}(|J|<k\/2)={\\mathbb{P}}(G_k^c)$ in the last.\n\nFor any $J\\in \\{0,1\\}^k,$ we let $\\eta_J$ be a Poisson process on \n${\\mathbb{R}}^d$ of rate 1, \nconditioned on the event $\\CD_J.$ Furthermore, for $j\\in J^c,$ let \n\\[\nD_j:=A_j \\setminus \\bigcup_{i\\in J} A_i.\n\\]\nWe see that $\\eta_J$ can be expressed as the sum of $\\tilde{\\eta}$ \nand $\\eta_D$ where $\\tilde{\\eta}$ is a Poisson process on \n${\\mathbb{R}}^d \\setminus \\bigcup_{n=1}^k A_n$ of unit rate, while $\\eta_D$ is \na Poisson process on $\\bigcup_{j\\in J^c} D_j$ conditioned on the \nevent $\\bigcap_{j\\in J^c} \\{\\eta(D_j)\\geq 1\\}.$ We let\n$C_1,\\ldots,C_m\\in {\\mathcal{B}}_\\alpha$ be a partition (up to sets of measure zero) of $\\bigcup_{j\\in J^c} D_j$,\nand use Lemma \\ref{lemma:elemPoisdom2} to see that \n$\\eta_D\\preceq \\eta_D'+\\xi_D$. Here of course, $\\eta'_D$ is Poisson process\non $\\bigcup_{j\\in J^c} D_j$ while $\\xi_D$ is a point process consisting of \none point added uniformly and independently to every box \n$C_i$ for $i=1,\\ldots,m.$ As in Lemma \\ref{lemma:elemPoisdom2} \n$\\eta'_D$ and $\\xi_D$ are independent.\n\nWe conclude that \n$\\eta_J=\\eta_D+\\tilde{\\eta} \\preceq \\eta'_D+\\xi_D+\\tilde{\\eta}$,\nand observe that $\\eta'_D+\\tilde{\\eta}$ is just a homogeneous Poisson\nprocess on ${\\mathbb{R}}^d\\setminus A_J$ where $A_J:=\\bigcup_{j\\in J} A_j.$\nWriting $\\eta_{A_J^c}$ for this sum, we have that \n$\\eta_J\\preceq \\xi_D+\\eta_{A_J^c}$ and by\nfirst using this, and then Markov's inequality, we get that\n\\begin{eqnarray} \\label{eqn:Markov}\n\\lefteqn{{\\mathbb{P}}(\\tilde{\\Psi}(y_c(B_1))\\geq h,\\ldots,\\tilde{\\Psi}(y_c(B_k))\\geq h\n|\\CD_J)}\\\\\n& & \\leq {\\mathbb{P}}_{\\xi_D+\\eta_{A_J^c}}\n(\\tilde{\\Psi}(y_c(B_1))\\geq h,\\ldots,\\tilde{\\Psi}(y_c(B_k))\\geq h) \\nonumber \\\\\n& & \\leq {\\mathbb{P}}_{\\xi_D+\\eta_{A_J^c}}\n\\left(s\\sum_{j\\in J}\\tilde{\\Psi}(y_c(B_j))\\geq h|J|s\\right)\n\\leq e^{-h|J|s}{\\mathbb{E}}_{\\xi_D+\\eta_{A_J^c}}\n\\left[e^{s\\sum_{j\\in J}\\tilde{\\Psi}(y_c(B_j))}\\right], \\nonumber\n\\end{eqnarray}\nwhere ${\\mathbb{P}}_{\\xi_D+\\eta_{A_J^c}}$ is the probability measure \ncorresponding to $\\xi_D+\\eta_{A_J^c},$ and where \n${\\mathbb{E}}_{\\xi_D+\\eta_{A_J^c}}$ denotes expectation with \nrespect to this probability measure. We have\n\\begin{eqnarray*}\n\\lefteqn{\\tilde{\\Psi}(y_c(B_j))=\\sum_{x\\in\\xi_D+\\eta_{A_J^c}}\n\\tilde{l}_\\alpha(|y_{c}(B_j)-x|)}\\\\\n& & =\\sum_{x\\in \\eta_{A_J^c}}\\tilde{l}_\\alpha(|y_{c}(B_j)-x|)\n+\\sum_{x\\in \\xi_D}\\tilde{l}_\\alpha(|y_{c}(B_j)-x|) \\\\\n& & =\\tilde{\\Psi}_{\\eta_{A_J^c}}(y_c(B_j))\n+\\tilde{\\Psi}_{\\xi_D}(y_c(B_j)),\n\\end{eqnarray*}\nusing obvious notation. Thus, using independence, we have that \n\\begin{equation} \\label{eqn:etaxi}\n{\\mathbb{E}}_{\\xi_D+\\eta_{A_J^c}}\n\\left[e^{s\\sum_{j\\in J}\\tilde{\\Psi}(y_c(B_j))}\\right]\n={\\mathbb{E}}_{\\eta_{A_J^c}}\n\\left[e^{s\\sum_{j\\in J}\\tilde{\\Psi}_{\\eta_{A_J^c}}(y_c(B_j))}\\right]\n{\\mathbb{E}}_{\\xi_D}\\left[e^{s\\sum_{j\\in J}\\tilde{\\Psi}_{\\xi_D}(y_c(B_j))}\\right].\n\\end{equation}\nConsider the function $g_J(y):=\\sum_{j\\in J} \\tilde{l}_\\alpha(|y_{c}(B_j)-y|),$ \nso that\n\\[\n\\sum_{x\\in \\eta_{A_J^c}} g_J(x)\n=\\sum_{x\\in \\eta_{A_J^c}}\\sum_{j\\in J} \\tilde{l}_\\alpha(|y_{c}(B_j)-x|)\n=\\sum_{j\\in J}\\sum_{x\\in \\eta_{A_J^c}}\\tilde{l}_\\alpha(|y_{c}(B_j)-x|)\n=\\sum_{j\\in J}\\tilde{\\Psi}_{\\eta_{A_J^c}}(y_{c}(B_j)),\n\\]\nand similarly \n\\[\n\\sum_{x\\in \\xi_D} g_J(x)=\\sum_{j\\in J}\\tilde{\\Psi}_{\\xi_D}(y_{c}(B_j)).\n\\]\nBy Campbell's theorem (see \\cite{Kingman} p 57-58, \\cite{D-VJ} Sections 2.4 and 9.4)\nwe get that\n\\begin{equation} \\label{eqn:cambeta}\n{\\mathbb{E}}_{\\eta_{A_J^c}}\n\\left[e^{s\\sum_{j\\in J} \\tilde{\\Psi}_{\\eta'_D+\\tilde{\\eta}}(y_{c}(B_j))}\\right]\n=\\exp\\left({\\int_{{\\mathbb{R}}^d\\setminus A_J}e^{sg_J(x)}-1 dx}\\right).\n\\end{equation}\nWe proceed by bounding the right hand side of this expression and \nstart by noting that for $x\\in {\\mathbb{R}}^d\\setminus A_J,$\n\\[\ng_J(x)\\leq \\tilde{\\Psi}_{A_o}(o)\\leq C I_\\alpha \\alpha^{-d}\n\\]\nso that by Lemma \\ref{lemma:bndedfield}, $g_J(x)$ is uniformly \nbounded by $CI_\\alpha\/\\alpha^d$ where $CI_\\alpha<\\infty$ is as in \nthat lemma. We can therefore use that $e^x-1\\leq 2x$ \nfor $x\\leq 1,$ to see that for $0<s\\leq \\alpha^d\/(CI_\\alpha),$ we have\n$e^{sg_J(x)}-1\\leq 2sg_J(x).$\nHence,\n\\begin{eqnarray} \\label{eqn:gboundeta}\n\\lefteqn{\\int_{{\\mathbb{R}}^d\\setminus A_J}e^{sg_J(x)}-1 dx \n\\leq \\int_{{\\mathbb{R}}^d\\setminus A_J}2sg_J(x) dx}\\\\\n& & = 2s\\int_{{\\mathbb{R}}^d\\setminus A_J}\\sum_{j\\in J} \n\\tilde{l}_\\alpha(|y_{c}(B_j)-x|) dx\n\\leq 2s\\sum_{j\\in J}\\int_{{\\mathbb{R}}^d\\setminus A(B_j)}\n\\tilde{l}_\\alpha(|y_{c}(B_j)-x|) dx \\nonumber \n=2sD|J|, \\nonumber\n\\end{eqnarray}\nwhere by using that the side length of $A(B_j)$ is \n$\\alpha(4\\lceil\\sqrt{d}\\rceil+1)$, we have that \n\\begin{eqnarray*}\n\\lefteqn{D=\\int_{{\\mathbb{R}}^d\\setminus A(B_j)}\\tilde{l}_\\alpha(|y_{c}(B_j)-x|) dx}\\\\\n& & =\\int_{{\\mathbb{R}}^d\\setminus A_o}\\tilde{l}_\\alpha(|x|)dx\n\\leq \\int_{{\\mathbb{R}}^d\\setminus A_o}\\tilde{l}_\\alpha(|x|-\\alpha \\sqrt{d})dx\n\\leq CI_\\alpha\n\\end{eqnarray*}\nas in the proof of Lemma \\ref{lemma:bndedfield}. \nUsing \\eqref{eqn:gboundeta} in \\eqref{eqn:cambeta}, we get that \n\\begin{equation} \\label{eqn:expetaest}\n{\\mathbb{E}}_{\\eta_{A_J^c}}\\left[e^{s\\sum_{j\\in J} \n\\tilde{\\Psi}_{\\eta_{A_J^c}}(y_{c}(B_j))}\\right]\n\\leq e^{2sCI_\\alpha|J|}.\n\\end{equation}\n\nWe now turn to the second factor on the right hand side of \\eqref{eqn:etaxi}.\nObserve that for any $k,$ $J\\in \\{0,1\\}^k$ and $j\\in J$ we have that \n\\[\n\\tilde{\\Psi}_{\\xi_D}(y_{c}(B_j))\\leq \\tilde{\\Psi}_{A_o}(o).\n\\]\nUsing Lemma \\ref{lemma:bndedfield}, it follows that \n\\begin{equation} \\label{eqn:expxiest}\n{\\mathbb{E}}_{\\xi_D}\\left[e^{s\\sum_{j\\in J} \\tilde{\\Psi}_{\\xi_D}(y_{c}(B_j))}\\right]\n\\leq e^{2sCI_\\alpha|J|\/\\alpha^d}.\n\\end{equation}\nUsing \\eqref{eqn:expetaest} and \\eqref{eqn:expxiest}, with \\eqref{eqn:etaxi}\nwe see from \\eqref{eqn:Markov} that \n\\[\n{\\mathbb{P}}(\\tilde{\\Psi}(y_c(B_1))\\geq h,\\ldots,\\tilde{\\Psi}(y_c(B_k))\\geq h|\\CD_J)\n\\leq \ne^{-h|J|s}e^{2sCI_\\alpha|J|}e^{2s|J|CI_\\alpha \/\\alpha^d}.\n\\]\nInserting this into \\eqref{eqn:prelest}\n\\begin{eqnarray*} \n\\lefteqn{{\\mathbb{P}}(\\sup_{y\\in B_1}\\Psi(y)\\geq h, \\ldots, \\sup_{y\\in B_k}\\Psi(y)\\geq h)}\\\\\n& & \\leq \\sum_{|J|\\geq k\/2}\ne^{-h|J|s}e^{2sCI_\\alpha|J|}e^{2s|J|CI_\\alpha\/\\alpha^d}\n{\\mathbb{P}}(\\CD_J)+e^{-d(\\alpha)k} \\nonumber \\\\\n& & \\leq e^{-C'I_\\alpha hsk\/\\alpha^d}+e^{-d(\\alpha)k},\n\\end{eqnarray*}\nfor some $C'>0.$ Finally, by first letting $\\alpha$ be so small \nthat $e^{-d(\\alpha)}\\leq \\epsilon\/2,$\nand then taking $h$ large enough, \\eqref{eqn:epsest} follows.\n\\fbox{}\\\\\n\nWe will now prove Theorem \\ref{thm:hcfinite} in its entirety. \\\\\n\n\\medskip\n{\\bf Proof of Theorem \\ref{thm:hcfinite}.}\nThe first statement is simply Theorem \\ref{thm:hcfinite_aux} and so we \nturn to the second statement.\n\nConsider the auxiliary attenuation function $l'(r):=l(r+1)$, and let\n$\\Psi'$ denote the corresponding random field. We observe that \nfor any $y\\in B(o,1)$ and $x\\in {\\mathbb{R}}^d,$\n$l'(|x|)=l(|x|+1)\\leq l(|x-y|-|y|+1)\\leq l(|x-y|),$\nso that \n\\[\n\\Psi'(o)=\\sum_{x\\in \\eta}l'(|x|)\n\\leq \\inf_{y\\in B(o,1)}\\sum_{x\\in \\eta}l(|x-y|)\n=\\inf_{y\\in B(o,1)}\\Psi(y).\n\\] \nWe proceed to show that ${\\mathbb{P}}(\\Psi'(o)=\\infty)=1,$ since then it follows that \n${\\mathbb{P}}\\left(\\inf_{y\\in {\\mathbb{R}}^d}\\Psi(y)=\\infty\\right)=1$ by a standard \ncountability argument. Therefore, let $A_0:=B(o,1),$ and \n$A_k:=B(o,k+1)\\setminus B(o,k)$ and note that \n$Vol(A_k)=\\kappa_d((k+1)^d-k^d)\\geq d\\kappa_d k^{d-1},$\nwhere $\\kappa_d$ denotes the volume of the unit ball in dimension $d.$\nFurthermore, let ${\\mathcal{A}}_k$ denote the event that $\\eta(A_k)\\geq \\kappa_d k^{d-1}.$\nFor any $X\\sim Poi(\\lambda),$ a standard Chernoff type bound yields\n\\[\n{\\mathbb{P}}(X\\leq \\lambda\/2)\\leq \\frac{e^{-\\lambda}\\left(e\\lambda\\right)^{\\lambda\/2}}\n{\\left(\\lambda\/2\\right)^{\\lambda\/2}}\n=\\left(\\frac{e}{2}\\right)^{-\\lambda\/2}=e^{-c\\lambda},\n\\]\nfor some $c>0.$ Therefore, ${\\mathbb{P}}({\\mathcal{A}}_k^c)\\leq e^{-cd\\kappa_d k^{d-1}}$ so that \n${\\mathbb{P}}({\\mathcal{A}}_k^c \\textrm{ i.o.})=0$ by the Borell-Cantelli lemma.\nThus, for a.e. $\\eta,$ there exists a $K=K(\\eta)<\\infty,$ so that \n${\\mathcal{A}}_k$ occurs for every $k\\geq K.$ Furthermore, we have that \nif ${\\mathcal{A}}_k$ occurs, then for any $k\\geq 3,$\n\\begin{eqnarray*}\n\\lefteqn{\\sum_{x\\in \\eta(A_k)}l'(|x|)\\geq \\kappa_d k^{d-1}l'(k+1)}\\\\\n& & =\\kappa_d k^{d-1}l(k+2)\n\\geq \\kappa_d \\frac{k^{d-1}}{(k+3)^{d-1}}\\int_{k+2}^{k+3}r^{d-1}l(r)dr\n\\geq \\frac{\\kappa_d}{2}\\int_{k+2}^{k+3}r^{d-1}l(r)dr.\n\\end{eqnarray*}\nTherefore we get that by letting $K\\geq 3,$\n\\[\n\\Psi'(o)=\\sum_{x\\in \\eta}l'(|x|)\n\\geq \\sum_{k=K}^\\infty \\frac{\\kappa_d}{2}\\int_{k+2}^{k+3}r^{d-1}l(r)dr\n=\\frac{\\kappa_d}{2}\\int_{K+2}^{\\infty}r^{d-1}l(r)dr=\\infty.\n\\]\n\\fbox{}\\\\\n\n\\section{Continuity of the field $\\Psi$} \\label{sec:cont}\n\nIn this section we will prove Proposition\n\\ref{prop:contfield}.\nWe will often use the following well known equality (see for instance \\cite{Kingman} p. 28)\n\\begin{equation} \\label{eqn:VanCamp}\n{\\mathbb{E}}\\left[\\sum_{x\\in \\eta} g(x)\\right]=\\int_{{\\mathbb{R}}^d} g(x)\\mu(dx),\n\\end{equation}\nwhere $\\eta$ is a Poisson process in ${\\mathbb{R}}^d$ with intensity measure $\\mu.$ \\\\\n\n\\noindent\n{\\bf Proof of Proposition \\ref{prop:contfield}.}\nWe start by proving the statement in the case when $l$ is bounded.\nFix $\\alpha,\\epsilon>0$, let $g_{y,z}(x)=|l(|x-y|)-l(|x-z|)|,$  \nand let $\\{D_n\\}_{n\\geq 1}$ be a sequence of bounded subsets of ${\\mathbb{R}}^d$\nsuch that $D_n \\uparrow {\\mathbb{R}}^d.$\nObserve that for any $\\delta>0,$\n\\begin{eqnarray} \\label{eqn:alphaprel}\n\\lefteqn{{\\mathbb{P}}\\left(\\sup_{y,z\\in B(o,1):|y-z|<\\delta} |\\Psi(y)-\\Psi(z)|\\geq \\epsilon\\right)}\\\\\n& & \\leq \n{\\mathbb{P}}\\left(\\sup_{y,z\\in B(o,1):|y-z|<\\delta} \\sum_{x\\in \\eta} g_{y,z}(x)\\geq \\epsilon\\right) \\nonumber\\\\\n& & \\leq \n{\\mathbb{P}}\\left(\\sup_{y,z\\in B(o,1):|y-z|<\\delta} \\sum_{x\\in \\eta(D_n)} g_{y,z}(x)\\geq \\epsilon\/2\\right)\n+{\\mathbb{P}}\\left(\\sup_{y,z\\in B(o,1):|y-z|<\\delta} \\sum_{x\\in \\eta(D_n^c)} g_{y,z}(x)\\geq \\epsilon\/2\\right).\\nonumber\n\\end{eqnarray}\nWe will proceed by bounding the two terms on the right hand side of \\eqref{eqn:alphaprel}.\nConsider therefore the second term \n\\begin{eqnarray} \\label{eqn:alpha1}\n\\lefteqn{\n{\\mathbb{P}}\\left(\\sup_{y,z\\in B(o,1):|y-z|<\\delta} \\sum_{x\\in \\eta(D_n^c)} g_{y,z}(x)\\geq \\epsilon\/2\\right)}\\\\\n& & \\leq {\\mathbb{P}}\\left(\\sup_{y,z\\in B(o,1):|y-z|<\\delta} \\sum_{x\\in \\eta(D_n^c)} l(|x-y|)+l(|x-z|)\\geq \\epsilon\/2\\right) \\nonumber\\\\\n& & \\leq {\\mathbb{P}}\\left(\\sup_{y\\in B(o,1)} \\sum_{x\\in \\eta(D_n^c)} l(|x-y|)+\n\\sup_{z\\in B(o,1)} \\sum_{x\\in \\eta(D_n^c)}l(|x-z|)\\geq \\epsilon\/2\\right) \\nonumber\\\\\n& & ={\\mathbb{P}}\\left(\\sup_{y\\in B(o,1)} \\sum_{x\\in \\eta(D_n^c)} l(|x-y|)\\geq \\epsilon\/4\\right).\n\\nonumber\n\\end{eqnarray}\nFurthermore, we have that for any\n$\\epsilon>0,$\n\\begin{eqnarray} \\label{eqn:Dnlimit}\n\\lefteqn{{\\mathbb{P}}\\left(\\sup_{y\\in B(o,1)} \\sum_{x\\in \\eta(D_n^c)}l(|x-y|)\\geq \\epsilon\\right)\n\\leq \\frac{1}{\\epsilon}{\\mathbb{E}}\\left[\\sup_{y\\in B(o,1)} \\sum_{x\\in \\eta(D_n^c)}l(|x-y|)\\right]}\\\\\n& & \\leq \\frac{1}{\\epsilon}{\\mathbb{E}}\\left[ \\sum_{x\\in \\eta(D_n^c)}\\sup_{y\\in B(o,1)}l(|x-y|)\\right]\n=\\frac{1}{\\epsilon} \\int_{{\\mathbb{R}}^d\\setminus D_n} \\sup_{y\\in B(o,1)}l(|x-y|) dx \\nonumber \\\\\n& & \\leq \\frac{1}{\\epsilon} \\int_{{\\mathbb{R}}^d\\setminus D_n} l\\left(\\max(|x|-2,0)\\right) dx\n\\to 0 \\textrm{ as } n\\to \\infty, \\nonumber\n\\end{eqnarray}\nwhere we use \\eqref{eqn:VanCamp} in the equality and the fact \nthat the intensity measure of $\\eta(D_n^c)$ is Lebesgue measure outside\nof $D_n.$ We also use the integrability assumption \n$\\int_0^\\infty r^{d-1}l(r)dr<\\infty$ when taking the limit.\nBy combining \\eqref{eqn:alpha1} and \\eqref{eqn:Dnlimit}, we see that by taking \n$n$ large enough, the second term of \\eqref{eqn:alphaprel} is smaller than $\\alpha.$\n\nFor the first term, we get that \n\\begin{eqnarray} \\label{eqn:alpha2}\n\\lefteqn{\n{\\mathbb{P}}\\left(\\sup_{y,z\\in B(o,1):|y-z|<\\delta} \\sum_{x\\in \\eta(D_n)} g_{y,z}(x)\n\\geq \\epsilon\/2\\right)}\\\\\n& & \\leq \\frac{1}{\\epsilon}\n{\\mathbb{E}}\\left[ \\sum_{x\\in \\eta(D_n)}\\sup_{y,z\\in B(o,1):|y-z|<\\delta} |l(|x-y|)-l(|x-z|)|\\right]\n\\nonumber \\\\\n& & =\\frac{1}{\\epsilon}\n\\int_{D_n}\\sup_{y,z\\in B(o,1):|y-z|<\\delta} |l(|x-y|)-l(|x-z|)| dx.\n\\nonumber\n\\end{eqnarray}\nSince $D_n$ is bounded, we have that for any $x\\in D_n,$\n\\[\n\\sup_{y,z\\in B(o,1):|y-z|<\\delta} |l(|x-y|)-l(|x-z|)|\n\\leq \\sup_{(r_1,r_2)\\in E_n}(l(r_1)-l(r_2))\n\\]\nwhere $E_n=\\{(r_1,r_2)\\in {\\mathbb{R}}^2:0\\leq r_1<r_2\\leq 2{\\rm diam}(D_n),|r_1-r_2|<\\delta\\}$. Since \n$l(r)$ is continuous, it is uniformly continuous on $[0,2{\\rm diam}(D_n)]$.\nTherefore, for any fixed $n,$ the right hand side of \\eqref{eqn:alpha2} is \nsmaller than $\\alpha$ for $\\delta$ small enough.\n\nWe conclude that for any $\\epsilon,\\alpha>0,$ there exists \n$\\delta>0,$ small enough so that\n\\begin{equation} \\label{eqn:2alpha}\n{\\mathbb{P}}\\left(\\sup_{y,z\\in B(o,1):|y-z|<\\delta} |\\Psi(y)-\\Psi(z)|\\geq \\epsilon\\right)\n\\leq 2 \\alpha.\n\\end{equation}\nTo conclude the proof, assume that $\\Psi(y)$ is not a.s. \ncontinuous everywhere. Then, with positive probability, there\nexists $\\epsilon>0$ and a point $w\\in B(o,1\/2)$ such that for \nany $\\delta>0$\n\\[\n\\sup_{y:|y-w|<\\delta} |\\Psi(y)-\\Psi(w)|\\geq \\epsilon,\n\\]\ncontradicting \\eqref{eqn:2alpha}.\n\nWe now turn to the case where $l$ is unbounded.\nThen, for any $M<\\infty,$ we let $l_M(r)=\\min(l(r),M),$\nand define $\\Psi_M(y)$ to be the random field obtained by using $l_M$ instead of $l.$\nIf we let  \n\\[\nB_M(x)=\\{y\\in {\\mathbb{R}}^d:l(|x-y|)\\geq M\\},\n\\]\nwe see that $\\Psi_M(y)=\\Psi(y)$ for every $y\\in {\\mathbb{R}}^d\\setminus\\cup_{x\\in \\eta}B_M(x).$\nBy the first case, $\\Psi_M(y)$ is continuous everywhere,\nand so  $\\Psi(y)$ is continuous for any \n$y\\in {\\mathbb{R}}^d\\setminus\\cup_{x\\in \\eta}B_M(x).$\nSince $M<\\infty$ was arbitrary, the statement follows after observing that \n$\\lim_{y \\to x}\\Psi(y)=\\infty$ whenever $x\\in \\eta.$\n\\fbox{}\\\\\n\n\n\\section{Uniqueness} \\label{sec:uniqueness}\n\n\nIn this section we restrict ourselves to $d=2$. We will first \nconsider the case of $\\Psi_{> h}$, and then explain how the second\ncase of Theorem \\ref{thm:uniqueness} quickly follows from it. For convenience\nwe formulate the following separate statement.\n\n\\begin{theorem}\n\\label{groter}\nLet $h$ be such that $\\Psi_{> h}$ contains an unbounded component.\nIf $l$ is continuous and with unbounded support, then for $d=2$,\nthere is a unique such unbounded component.\n\\end{theorem} \nOur strategy will be to adapt the argument in \\cite{GKR} which proves uniqueness \nfor a class of models on ${\\mathbb{Z}}^d$. In order to perform this adaptation\nit is much easier to work with {\\em arcwise} connectedness rather than \nconnectedness. The reason for this is that we can easily form new arcs from intersecting \narcs, while the corresponding result for connectedness is rather challenging topologically.\n\n\nHowever, in our continuous context, we have defined percolation in terms of \nconnectedness, as is usually done. But, since $\\Psi$ is a.s.\\ continuous by Proposition \n\\ref{prop:contfield}, the set $\\Psi_{>h}$ is a.s.\\ an open set. For \nopen sets, connected and arcwise connected are the same thing, as is \nwell known. Hence, if $x$ and $y$ are in the same connected component \nof $\\Psi_{>h}$, then there is a continuous curve from $x$ to $y$ in \n$\\Psi_{>h}$. This observation makes $\\Psi_{>h}$ easier to study than $\\Psi_{\\geq h}$ \ndirectly, and is the reason for proving Theorem \\ref{groter} separately.\n\nIn the sequel we try to balance between \nthe fact that we do not want or need to repeat the whole argument of \n\\cite{GKR} on the \none hand, and the need to explain in detail what changes are to be made and \nwhat these changes constitute on the other hand.\n\nIn \\cite{GKR}, uniqueness of the infinite cluster in two-dimensional \ndiscrete site percolation is proved under four conditions. Consider a probability \nmeasure $\\mu$ on $\\{0,1\\}^{\\mathbb{Z}^2}$ and let \n$\\omega\\in \\{0,1\\}^{\\mathbb{Z}^2}$ be a configuration. If $\\omega(x)=1$\nwe call $x\\in {\\mathbb{Z}}^2$ {\\em open} and if $\\omega(x)=0$ we call it {\\em closed}.\nThe four conditions which together imply uniqueness of the infinite \nopen component are:\n\n\\begin{enumerate}\n\\item $\\mu$ is invariant under horizontal and vertical translations and under axis reflections.\n\\item $\\mu$ is ergodic (separately) under horizontal and vertical translations.\n\\item $\\mu(E \\cap F) \\geq \\mu(E)\\mu(F)$ for events $E$ and $F$ which are both increasing or both decreasing (The FKG inequality).\n\\item The probability that the origin is in an infinite cluster is non-trivial, that is, strictly between 0 and 1.\n\\end{enumerate}\n\nIn our context, conditions analogous to Conditions 1 and 2 clearly hold. Some \ncare is needed for Condition 3 though. We will say that an event $E$ is \nincreasing if a configuration in $E$ remains in $E$ if we add additional \npoints to the point process $\\eta$ (and adapt the field accordingly). \nFurthermore, $E$ is decreasing if $E^c$ is increasing. With these \ndefinitions, one can prove the analogue of the FKG inequality as in \nthe proof of Theorem 2.2 in \\cite{MR}.\n\nCondition 4 is natural in the discrete context. Indeed, if the probability that the origin is in an infinite cluster is 0, then by translation invariance, no infinite cluster can exist a.s. The case in which the probability that the origin is in an infinite cluster is 1 was excluded only for convenience, and this assumption is not used in the proof in \\cite{GKR}. In our continuous context, we need to be slightly more careful. Suppose that $\\Psi_{>h}$ contains an unbounded component with positive probability. Since $\\Psi_{>h}$ is an open set by continuity of the field, any such unbounded component must be open as well. Hence there must be an $\\epsilon>0$ and an $x \\in {\\mathbb{R}}^2$ so that $B(x,\\epsilon)$ is contained in an infinite component with positive probability, since a countable collection of such balls covers the plane. By translation invariance, the same must then be true for any $x\\in {\\mathbb{R}}^2$. Hence, any point $x \\in {\\mathbb{R}}^2$ is contained in an infinite component with positive probability, and Condition 4 holds. \n\nGandolfi, Keane and Russo prove uniqueness by showing that there exists a $\\delta>0$ such that any box $B_n=[-n,n]^2$ is surrounded by an open path with probability at least $\\delta$. Hence the probability that all such boxes are surrounded by an open path is at least $\\delta$, and since the latter event is translation invariant it must have probability one. This ensures uniqueness, as is well known since 1960, see \\cite{Harris}. For the proof of Theorem \\ref{groter}, we can \nin principle follow the structure of their argument, with a number \nof modifications, as follows.\n\n\n\n\n\n\\medskip\n\\noindent\n{\\bf Proof of Theorem \\ref{groter}.}\nFor any set $A\\subset {\\mathbb{R}}^d,$ we will let $\\Gamma_A:=\\sup_{x\\in A} \\Psi(x).$\n\nThe first step is to show that it is impossible to have percolation in a \nhorizontal strip $Q_M$ of the form\n$$\nQ_M: =\\{(x,y) \\in {\\mathbb{R}}^2; -M \\leq y \\leq M\\},\n$$\nand similarly for vertical strips. In their case this claim simply follows \nfrom the fact that closed sites exist (by virtue of Condition 4) and \nthen it follows from  Condition 3 that there is a positive probability \nthat a strip is blocked completely by closed sites. Finally, ergodicity \n(or rather stationarity) shows that a strip is blocked infinitely many \ntimes in either direction.\n\nSince we work in a continuous setting, this argument does not go through \nimmediately. However, we can adapt it to our context. To this end, consider the \nset $C= C(N,M):=[N,N+1] \\times [-M, M]$, that is, a vertical ``strip\" in $Q_M$.  \nSince the field is a.s.\\ finite, by deleting points one by one from $\\eta$, \nsay in increasing order with respect to distance to $C$, we have that \nupon deleting these points, $\\Gamma_C \\downarrow 0$. Hence, after \ndeleting sufficiently many points it must be the case that \n$\\Gamma_C <h$, for any given $h >0$. If we let $\\CD^o(L)$\ndenote the event that the contribution of points outside the \nbox $B_{L}$ to $\\Gamma_C$ is at most $h$, we conclude that for some\n(random) number $L$, $\\CD^o(L)$ occurs. It then follows that \nfor some deterministic $L_0,$ ${\\mathbb{P}}(\\CD^o(L_0))>0.$ Note also that \n$\\CD^o(L_0)$ only depends on the points of $\\eta$ outside $B_{L_0}$. \n\nLet $\\CD^i(L_0)$ denote the event that there are no points of $\\eta$\nin $B_{L_0}$ itself. Then ${\\mathbb{P}}(\\CD^i(L_0))>0$ and by independence of\n$\\CD^i(L_0)$ and $\\CD^o(L_0)$, it also follows that \n${\\mathbb{P}}(\\CD^i(L_0) \\cap \\CD^o(L_0))>0$. Furthermore, on the event  \n$\\CD^i(L_0) \\cap \\CD^o(L_0),$ we have that $\\Gamma_C <h$, and \nwe conclude that for any $h>0$ there is positive probability that \nthe field $\\Psi$ does not exceed $h$ on $C$. \nSo any vertical strip in $C(N,M) \\subset Q_M$ has positive \nprobability to satisfy $\\Gamma_C <h$ and by stationarity there must \nbe infinitely many of such strips in both directions. So there \ncan not be percolation in $Q_M$. \n\n\\medskip\n\nHaving established this, we now turn to the second step of the argument.\nAs mentioned above, in \\cite{GKR}, they construct open paths whose union, by virtue of \ntheir specific construction, surround a given box. They show that \nfor any given box, such a construction can be carried out with a \nuniform positive probability. It is at this point of the argument \nthat two-dimensionality is crucial as the two-dimensional topology \nforces certain paths to intersect.\n\nThe remainder of the proof of uniqueness proceeds in two steps. \nFirst we prove uniqueness under the assumption that the half-plane \n$H^+:=\\{(x,y); y \\geq 0\\}$ percolates. After that, \nwe prove uniqueness under the assumption that $H^+$ does not percolate. \n\nFor $x \\in {\\mathbb{R}}^2, A, B \\subset {\\mathbb{R}}^2$, we write $E(x, A, B)$ for the event \nthat there is a continuous path in  $\\Psi_{>h}$ from $x$ to $A$ which is \ncontained in $B$, and $E(x, \\infty, B)$ if there is an unbounded \ncontinuous path from $x$ in $B$. We write $L_N:=\\{(x,y); y=N\\}$,  \n$L_N^+:=\\{(x,y); y=N, x \\geq 0\\}$ and  $L_N^-:=\\{(x,y); y=N, x \\leq 0\\}$. \nFinally we write $H_N^+:=\\{(x,y); y \\geq N\\}$, so that $H_0^+ = H^+$.\n\nLet $E :=E(0,\\infty, H^+)$, let $D$ be a box centered at the origin, and \nlet $D_N := D + (0,N)$. \nFinally, let $\\tilde{E}_N := E(0, \\infty, H^+ \\backslash D_N)$. Now,\n\\[\n{\\mathbb{P}}(E){\\mathbb{P}}(\\Gamma_{D}) \\geq {\\mathbb{P}}(\\tilde{E}_N){\\mathbb{P}}(\\Gamma_{D_N}) \n\\geq {\\mathbb{P}}(\\tilde{E}_N \\cap \\Gamma_{D_N})= {\\mathbb{P}}(E \\cap  \\Gamma_{D_N}).\n\\]\nSince our system is mixing (see e.g.\\ \\cite{MR} p. 26\nplus the fact that the field is a deterministic function of the Poisson process), \nwe have that $\\lim_{N \\to \\infty} {\\mathbb{P}}(E \\cap  \\Gamma_{D_N})\n={\\mathbb{P}}(E){\\mathbb{P}}(\\Gamma_{D})$. It follows that when $N \\to \\infty$, ${\\mathbb{P}}(\\tilde{E}_N) \\to {\\mathbb{P}}(E)$. \nIn words, if we have percolation from the origin in $H^+$, the conditional probability that \nthere is an unbounded path avoiding $D_N$ tends to 1 as $N \\to \\infty$.\n\nHence, if the probability that $y_{-N}:=(0,-N)$ percolates in $H^+_{-N}$ is  $\\delta >0$, then\nfor $N$ large enough,\n$$\n{\\mathbb{P}}(E(y_{-N}, \\infty, H_{-N}^+\\backslash D))\\geq \\delta\/2.\n$$ \nSince the strip $Q_N$ does not percolate, if $y_{-N}$ percolates in \n$H^+_{-N}\\backslash D$, we conclude that the event \n$E(y_{-N}, L_N, Q_N\\backslash D)$ must occur, so that \n${\\mathbb{P}}(E(y_{-N}, L_N, Q_N\\backslash D)) \\geq \\delta\/2$. \n\nThe endpoint of the curve in the event $E(y_{-N}, L_N, Q_N\\backslash D)$\nis either in $L_N^+$ or in $L_N^-$, and by reflection symmetry, both \noptions have the same probability. Hence, \n$$\n{\\mathbb{P}}(E(y_{-N}, L_N^+, Q_N\\backslash D))\\geq \\delta\/4.\n$$\nBy reflection symmetry, it then follows that also\n$$\n{\\mathbb{P}}(E(y_{N}, L_{-N}^+, Q_N\\backslash D))\\geq \\delta\/4,\n$$\nand by combining the curves in the last two displayed formulas and the \nFKG inequality, we find that\n\\begin{equation}\n\\label{curves}\n{\\mathbb{P}}(E(y_{N}, y_{-N}, Q_N\\backslash D))\\geq \\delta^2\/16.\n\\end{equation}\nAny curve in the event $E(y_{N}, y_{-N}, Q_N\\backslash D)$\neither has $D$ on the left or on the right (depending whether it has \npositive or negative winding number) and again by reflection symmetry, \nboth possibilities must have probability at least $\\delta^2\/32$.  \nLet $J^+$ ($J^-$) be the sub-event of $E(y_{N}, y_{-N}, Q_N\\backslash D)$\nwhere there exists a curve with positive (negative) winding number.\nBy the FKG inequality, we \nhave that ${\\mathbb{P}}(J^+ \\cap J^-) \\geq \\delta^4\/1024$. But on $J^+ \\cap J^-$, \nthe box $D$ is surrounded by a continuous curve in $\\Psi_{>h}$, and we are done. \n\nFinally, we consider the case in which the half space does not percolate. \nWe can modify the argument in \\cite{GKR} similarly and we do not spell out \nall details. In the first case we showed that if we have percolation from \nthe origin in $H^+$, the conditional probability that there is an unbounded \npath avoiding $D_N$ tends to 1 as $N \\to \\infty$. In this second case it \nturns out that we need to show that this remains true if we in addition \nalso want to avoid $D_{-N}$. For this, the usual mixing property that we \nused above, does not suffice, and a version of 3-mixing is necessary. As \nin \\cite{GKR}, we use Theorem 4.11 in \\cite{Fu} for this, in which it is \nshown that ordinary weak mixing implies 3-mixing along a sequence of density 1. \nSince our system is weakly mixing, this application of Theorem 4.11 in \n\\cite{Fu} is somewhat simpler than in \\cite{GKR}, but other than that \nour argument is the same, and we do not repeat it here.\n\\fbox{}\\\\\n\n\n\\medskip\nFinally we show how Theorem \\ref{groter} implies Theorem \\ref{thm:uniqueness}.\n\n\\medskip\n\\noindent\n{\\bf Proof of Theorem \\ref{thm:uniqueness}}.\nWe first claim that $\\Psi_{>h}$ percolates if and only if $\\Psi_{\\geq h}$ percolates. The ``only if\" is \nclear, since $\\Psi_{> h} \\subset \\Psi_{\\geq h}$. \n\nNext, suppose that $\\Psi_{\\geq h}$ percolates. By definition, this implies that $\\Psi_{\\geq h}$ a.s.\\ contains an unbounded connected component. Let us denote this event by $C_h$. Let $A$ be a bounded region with positive volume. Since the probability of $C_h$ is 1, it must be the case that \n$$\n{\\mathbb{P}}(C_h| \\eta(A)=0)=1,\n$$\nwhere $\\eta(A)$ is the number of points of the Poisson process in $A$.\nSince we can sample from the conditional distribution of the process given $\\eta(A)=0$ by first sampling unconditionally and then simply remove all points in $A$, it follows that we cannot destroy the event of percolation by removing all points in $A$.\n\nHence if we take all points out from $A$, the resulting field $\\Psi^A$, say, will be such that $\\Psi^A_{\\geq h}$ percolates a.s. But if $\\eta(A) >0$, then it is the case that \n$\\Psi^A_{\\geq h} \\subseteq \\Psi_{>h}$, and it is precisely here we assume that \nthe attenuation function $l$ has unbounded support. Hence, with positive probability we have that $\\Psi_{>h}$ percolates, and by ergodicity this implies that $\\Psi_{>h}$ contains an infinite component a.s.\n\nWe can now quickly finish the proof. Suppose that $\\Psi_{\\geq h}$ percolates. Then, as we just saw, also \n$\\Psi_{>h}$ percolates. Hence we can apply the proof of Theorem \\ref{groter}, and conclude that \n$\\Psi_{>h}$ does contain continuous curves around each box. Since $\\Psi_{\\geq h}$ is an even larger set, the \nsame must be true for $\\Psi_{\\geq h}$ and uniqueness for this latter set follows as before.\n\\fbox{}\\\\\n\n\\noindent \n{\\bf Remark:}In light of Theorem \\ref{thm:uniqueness}, it is of course natural to expect \nthat uniqueness should hold also for $d\\geq 3.$ The classical argument for uniqueness\nin various lattice models and continuum percolation consists of two parts. Below\nwe examine these separately.\n\n\nLet $N_h$ be the number of unbounded components in $\\Psi_{\\geq h}$. \nFollowing the arguments of \\cite{NS} (which is for the lattice case \nbut can easily be adapted to the setting of Boolean percolation, see \n\\cite{MR} Proposition 3.3) one starts by observing that\n${\\mathbb{P}}(N_h=k)=1$ for some $k\\in\\{0,1,\\ldots\\}\\cup\\{\\infty\\}.$  \nAssume for instance that ${\\mathbb{P}}(N_h=3)=1$, and proceed by taking a \nbox $[-n,n]^d$ large enough so that \nat least two of these\ninfinite components intersect the box with positive probability. Then, glue these \ntwo components together by the use of a finite energy argument. That is, turn all \nsites in the box to state $1$ in the discrete case, or add balls to \nthe box in the \nBoolean percolation case. In this way, we reduce $N_h$ by (at least) 1, \nshowing that \n${\\mathbb{P}}(N_h=3)<1,$ a contradiction. If one attempts to repeat this procedure \nin our setting (with the support of $l$ being unbounded), \none finds that by adding points to the field, the gluing of two \ninfinite components might at the same\ntime result in the forming of a completely new infinite component \nsomewhere outside the box. \nTherefore, one cannot conclude that ${\\mathbb{P}}(N_h=3)<1$.\n\nThe second difficulty occurs when attempting to rule out the possibility that \n${\\mathbb{P}}(N_h=\\infty)=1.$ For the Boolean percolation model one uses an argument by Burton and Keane in \\cite{BK},\nadapted to the case of Boolean percolation (see \\cite{MR} proof of Theorem 3.6). \nHowever this argument hinges on the trivial\nbut crucial fact that for this model any unbounded component must contain infinitely many \npoints of the Poisson process $\\eta$. This is not the case in our setting. An unbounded component\ncan in principle contain only a finite number of points of $\\eta$, \nor indeed none at all. \\\\\n\n\nWe now turn to the last result of this paper, Theorem \\ref{thm:thetacont}.\nIn order to prove continuity, we will give separate arguments for \nleft- and right-continuity. The strategy to prove right-continuity \nwill be similar to the corresponding result (i.e. left-continuity)\nfor discrete lattice percolation (see \\cite{Grimmett}, Section 8.3).\nHowever, while the other case is trivial for discrete percolation, \nthis is where most of the effort in proving Theorem \\ref{thm:thetacont}\nlies. Before giving the full proof, we will need to establish two \nlemmas that will \nbe used to prove left-continuity. See also the remark after the end of \nthe proof of Theorem \\ref{thm:thetacont}.\n\n\nLet $X^0\\sim$Poi$(\\lambda)$, and let $X^1=X^0+1.$ The following\nlemma provides a useful coupling.\n\\begin{lemma} \\label{lemma:Poicouple}\nThere exist random variables $Y^0\\stackrel{d}{=}X^0$ and \n$Y^1\\stackrel{d}{=}X^1$ coupled so that \n\\[\n{\\mathbb{P}}(Y^0\\neq Y^1)\n=\\frac{\\lambda^{\\lfloor \\lambda \\rfloor}+1}{(\\lfloor \\lambda \\rfloor+1)!}\ne^{-\\lambda}.\n\\]\n\\end{lemma}\n\\noindent\n{\\bf Proof.}\nIn what follows, sums of the form $\\sum_{l=M}^{M-1} a_l$ is understood to be 0,\nand in order not to introduce cumbersome notation, expressions such as\n$\\lambda^k\/k!$ will be interpreted as 0 for $k<0.$\nNote also that \n\\begin{equation} \\label{eqn:Poiratio}\n\\frac{{\\mathbb{P}}(X^0=k)}{{\\mathbb{P}}(X^1=k)}\n=\\frac{\\frac{\\lambda^k}{k!}e^{-\\lambda}}\n{\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda}}\n=\\frac{\\lambda}{k}\\geq 1 \\textrm{ iff } k\\leq \\lambda.\n\\end{equation}\n\n\nWe start by giving the coupling and then verify that it is well defined\nand has the correct properties.\nLet $U \\sim U[0,1]$ and for $1\\leq k \\leq \\lambda$ let $Y^0=Y^1=k$ if\n\\begin{equation} \\label{eqn:Poi1}\n\\sum_{l=0}^{k-2}\\frac{\\lambda^{l}}{l!}e^{-\\lambda}\n<U\\leq \\sum_{l=0}^{k-1}\\frac{\\lambda^{l}}{l!}e^{-\\lambda}\n\\end{equation}\nwhile for $k>\\lambda$ we let $Y^0=Y^1=k$ if\n\\begin{equation} \\label{eqn:Poi2}\n\\sum_{l=0}^{\\lfloor\\lambda\\rfloor}\\frac{\\lambda^l}{l!}e^{-\\lambda}\n+\\sum_{l=\\lfloor\\lambda\\rfloor +2}^{k-1}\\frac{\\lambda^l}{l!}e^{-\\lambda}\n<U\\leq \\sum_{l=0}^{\\lfloor\\lambda\\rfloor}\\frac{\\lambda^l}{l!}e^{-\\lambda}\n+\\sum_{l=\\lfloor\\lambda\\rfloor +2}^{k}\\frac{\\lambda^l}{l!}e^{-\\lambda}\n\\end{equation}\nFurthermore, we let $Y^0=k$ if $k\\leq \\lambda$ and\n\\begin{equation} \\label{eqn:Poi3}\n1-\\frac{\\lambda^k}{k!}e^{-\\lambda}\n< U\\leq 1-\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda}\n\\end{equation}\nwhile $Y^1=k$ if $k> \\lambda$ and\n\\begin{equation} \\label{eqn:Poi4}\n1-\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda}<U\n\\leq 1-\\frac{\\lambda^{k}}{k!}e^{-\\lambda}.\n\\end{equation}\nConsider now \\eqref{eqn:Poi3}. Since $k\\leq \\lambda$ it follows from \n\\eqref{eqn:Poiratio} that \n\\[\n1-\\frac{\\lambda^k}{k!}e^{-\\lambda} \n\\leq 1-\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda},\n\\]\nwith equality iff $k=\\lambda.$ It follows that \\eqref{eqn:Poi3}\nis well defined, and similarly we can verify that \\eqref{eqn:Poi4}\nis also well defined.\n\nWe proceed to verify that \\eqref{eqn:Poi1}--\\eqref{eqn:Poi4} \ngives the correct distributions of $Y^0$ and $Y^1$. \nTo this end, observe that from \\eqref{eqn:Poi1} and \\eqref{eqn:Poi3} \nwe have that for $0\\leq k\\leq \\lambda$ \n\\[\n{\\mathbb{P}}(Y^0=k)=\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda}\n+\\frac{\\lambda^{k}}{k!}e^{-\\lambda}-\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda}\n=\\frac{\\lambda^{k}}{k!}e^{-\\lambda}.\n\\]\nFurthermore, from \\eqref{eqn:Poi1} we get that for $k>\\lambda,$\n\\[\n{\\mathbb{P}}(Y^0=k)=\\frac{\\lambda^{k}}{k!}e^{-\\lambda}\n\\]\nso that indeed $Y^0\\sim$Poi$(\\lambda).$\n\n\nSimilarly, we see from \\eqref{eqn:Poi1} that for \n$1\\leq k \\leq \\lambda$ we have that \n\\[\n{\\mathbb{P}}(Y^1=k)=\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda},\n\\]\nwhile by summing the contributions from \\eqref{eqn:Poi2}\u00a8and \n\\eqref{eqn:Poi4} we get that for $k>\\lambda$\n\\[\n{\\mathbb{P}}(Y^1=k)\n=\\frac{\\lambda^k}{k!}e^{-\\lambda}+\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda}\n-\\frac{\\lambda^{k}}{k!}e^{-\\lambda}\n=\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda},\n\\]\nso that $Y^1$ has the desired distribution.\n\nFinally, the lemma follows by observing that \n\\[\n{\\mathbb{P}}(Y^0\\neq Y^1)={\\mathbb{P}}\\left(U>\\sum_{l=0}^{\\lfloor\\lambda\\rfloor}\\frac{\\lambda^l}{l!}e^{-\\lambda}\n+\\sum_{l=\\lfloor\\lambda\\rfloor+2}^{\\infty}\\frac{\\lambda^l}{l!}e^{-\\lambda}\\right)\n=\\frac{\\lambda^{\\lfloor \\lambda \\rfloor}+1}{(\\lfloor \\lambda \\rfloor+1)!}\ne^{-\\lambda}.\n\\]\n\\fbox{}\\\\\n\nLet $\\eta^0_n$ be a homogeneous Poisson process in ${\\mathbb{R}}^2$ with rate \n1, and let $\\eta^1_n$ be a point process such that \n$\\eta^1_n\\stackrel{d}{=} \\eta^0_n+\\delta_{V_n}$ where $V_n\\sim$U$(B_n)$\nand $B_n=[-n,n]^2$.\nThus $\\eta^1_n$ is constructed by adding a point uniformly located within \nthe box $B_n$ to a homogeneous Poisson process in ${\\mathbb{R}}^2$.\nLet ${\\mathbb{P}}_n^i$ be the distribution of $\\eta^i_n$ for $i=0,1$\nand let \n\\[\nd_{TV}({\\mathbb{P}}_n^0,{\\mathbb{P}}_n^1):=\\sup_{A}|{\\mathbb{P}}_n^0(A)-{\\mathbb{P}}_n^1(A)|\n\\]\nbe the total variation distance between ${\\mathbb{P}}_n^0$ and ${\\mathbb{P}}_n^1$, where the supremum\nis taken over all measurable events $A$.\n\n\\begin{lemma} \\label{lemma:PoiTV}\nFor any $n\\geq 1$ we have that \n\\[\nd_{TV}({\\mathbb{P}}_n^0,{\\mathbb{P}}_n^1)\\leq \\frac{(4n^2)^{4n^2+1}}{(4n^2+1)!}e^{-4n^2}\n\\leq n^{-1}.\n\\]\n\\end{lemma}\n\\noindent\n{\\bf Proof.}\nLet $\\lambda=4n^2$ and pick $Y^0,Y^1$ as in Lemma \\ref{lemma:Poicouple}.\nFurthermore, let $\\eta$ be a homogeneous Poisson process in ${\\mathbb{R}}^2,$\nindependent of $Y^0$ and $Y^1,$\nand let $(U_k)_{k\\geq 1}$ be an i.i.d. sequence independent of $Y^0,Y^1$\nand $\\eta$ and such that $U_k\\sim$U$(B_n)$. Then, define \n\\[\n\\eta_n^0:=\\eta(B_n^c)+\\sum_{k=1}^{Y^0}\\delta_{U_k},\n\\]\nand\n\\[\n\\eta_n^1:=\\eta(B_n^c)+\\sum_{k=1}^{Y^1}\\delta_{U_k}.\n\\]\nIt is easy to see that $\\eta_n^i\\sim {\\mathbb{P}}_n^i$ and that \n\\begin{equation} \\label{eqn:etaY}\n{\\mathbb{P}}(\\eta_n^0\\neq \\eta_n^1)={\\mathbb{P}}(Y^0\\neq Y^1).\n\\end{equation}\nThus, for any measurable event $A$, we have that \n\\[\n|{\\mathbb{P}}_n^0(A)-{\\mathbb{P}}_n^1(A)|={\\mathbb{P}}(\\eta_n^0\\in A, \\eta_n^1 \\not \\in A)\n+{\\mathbb{P}}(\\eta_n^1\\in A, \\eta_n^0 \\not \\in A)\\leq \n{\\mathbb{P}}(\\eta_n^0\\neq \\eta_n^1)\n\\leq \\frac{(4n^2)^{4n^2+1}}{(4n^2+1)!}e^{-4n^2},\n\\]\nby using \\eqref{eqn:etaY} and Lemma \\ref{lemma:Poicouple}.\n\nFurthermore, by Stirling's approximation, we see that \n\\[\n\\frac{(4n^2)^{4n^2+1}}{(4n^2+1)!}e^{-4n^2}\n\\leq \\frac{(4n^2)^{4n^2}}{4n^2!}e^{-4n^2}\n\\leq \\frac{(4n^2)^{4n^2}}{\\sqrt{2\\pi}(4n^2)^{4n^2+1\/2}e^{-4n^2}}e^{-4n^2}\n\\leq n^{-1}.\n\\]\n\\fbox{}\\\\\n\\noindent\n{\\bf Remark:} Although we choose to state and prove this only for $d=2,$\na version of this lemma obviously holds for all $d\\geq 1.$ \\\\\n\nWe are now ready to give the proof of our last result.\\\\\n\\noindent\n{\\bf Proof of Theorem \\ref{thm:thetacont}.} \nWe start by proving the left-continuity of $\\theta_{>}(h).$ \nWe claim that \n\\begin{equation} \\label{eqn:uplimg1}\n\\lim_{g \\uparrow h} \\theta_{>}(g)={\\mathbb{P}}({\\mathcal{C}}_{o,>}(g) \\textrm{ is unbounded for every }\ng<h)={\\mathbb{P}}({\\mathcal{C}}_{o,\\geq}(h) \\textrm{ is unbounded}).\n\\end{equation}\nTo see this, observe first that trivially \n\\[\n\\{{\\mathcal{C}}_{o,\\geq}(h) \\textrm{ is unbounded}\\} \\subset \n\\bigcap_{g<h} \\{{\\mathcal{C}}_{o,>}(g) \\textrm{ is unbounded}\\}.\n\\]\nSecondly, assume that ${\\mathcal{C}}_{o,\\geq}(h)$ is bounded. Since \n${\\mathcal{C}}_{o,\\geq}(h)$ and $\\Psi_{\\geq h} \\setminus {\\mathcal{C}}_{o,\\geq}(h)$\nare disconnected, there exist open sets $G_1,G_2$ such that $G_1$ is connected,\n${\\mathcal{C}}_{o,\\geq}(h) \\subset G_1$, $\\Psi_{\\geq h} \\setminus {\\mathcal{C}}_{o,\\geq}(h)\\subset G_2$\nand $G_1 \\cap G_2=\\emptyset.$ Therefore, the set $G_3=G_1\\setminus {\\mathcal{C}}_{o,\\geq}(h)$\nis an open connected set separating the origin $o$ from $\\infty.$ Since \n$G_3$ is then also arcwise connected, it follows that it \n must contain a circuit surrounding the origin.\nThat is, there exists a continuous function $\\gamma:[0,1] \\to {\\mathbb{R}}^2$ such \nthat $\\gamma(0)=\\gamma(1)$ and $\\gamma$ separates $o$ from $\\infty.$\nSince $\\gamma$ is continuous,\nthe image of $\\gamma$ (Im$(\\gamma)$) is compact, and so \n$\\sup_{t\\in[0,1]}\\Psi(\\gamma(t))$ is obtained, since $\\Psi$ is continuous\nby Proposition \\ref{prop:contfield}. By construction, $G_3 \\subset {\\mathbb{R}}^2\\setminus \n\\Psi_{\\geq h}$ and so \nIm$(\\gamma) \\subset {\\mathbb{R}}^2 \\setminus \\Psi_{\\geq h}.$ We conclude that \n$\\sup_{t\\in[0,1]}\\Psi(\\gamma(t))<h.$ Therefore, for any $g$ such that\n$\\sup_{t\\in[0,1]}\\Psi(\\gamma(t))<g<h$ we must have that ${\\mathcal{C}}_{o,>}(g)$ is bounded.\nThis proves \\eqref{eqn:uplimg1}.\n\n\nLet $n$ be any integer and take\n\\[\n\\eta\\in \\{{\\mathcal{C}}_{o,\\geq}(h) \\textrm{ is unbounded}\\} \\setminus \n\\{{\\mathcal{C}}_{o,>}(h) \\textrm{ is unbounded}\\}.\n\\]\nLet $\\eta^1_n=\\eta+\\delta_{V_n}$ where $V_n\\sim$U$(B_n)$ and observe \nthat since $l$ has unbounded support,\n\\[\n\\eta_n^1\\in \\{{\\mathcal{C}}_{o,>}(h) \\textrm{ is unbounded}\\}.\n\\]\nUsing Lemma \\ref{lemma:PoiTV} we get that \n\\begin{eqnarray*}\n\\lefteqn{{\\mathbb{P}}({\\mathcal{C}}_{o,\\geq}(h) \\textrm{ is unbounded})} \\\\\n& & \\leq {\\mathbb{P}}^1_n({\\mathcal{C}}_{o,>}(h) \\textrm{ is unbounded})\n\\leq {\\mathbb{P}}^0_n({\\mathcal{C}}_{o,>}(h) \\textrm{ is unbounded})+n^{-1}\n=\\theta_{>}(h)+n^{-1}.\n\\end{eqnarray*}\nThis together with \\eqref{eqn:uplimg1} yields\n\\[\n\\lim_{g \\uparrow h} \\theta_{>}(g)\\leq \\lim_{n \\to \\infty}\\theta_{>}(h)+n^{-1}\n=\\theta_{>}(h).\n\\]\n\n\nIt remains to prove that \n\\begin{equation} \\label{eqn:downlimg1}\n\\lim_{g \\downarrow h} \\theta_{>}(g)=\\theta_{>}(h),\n\\end{equation}\nfor $h<h_c,$ and we will use a similar approach as above.\n We note that \n\\begin{equation} \\label{eqn:downlimg2}\n\\lim_{g \\downarrow h} {\\mathbb{P}}({\\mathcal{C}}_{o,>}(g) \\textrm{ is unbounded})\n={\\mathbb{P}}({\\mathcal{C}}_{o,>}(g) \\textrm{ is unbounded for some } g>h).\n\\end{equation}\nAssume that ${\\mathcal{C}}_{o,>}(h)$ contains an unbounded component, and consider any \n$h<g<h_c.$ Since $\\Psi_{>g}$ also must contain an unbounded component $I_g,$\nand since by Theorem \\ref{thm:uniqueness} we know that this is unique, \nwe conclude that $I_g \\subset {\\mathcal{C}}_{o,>}(h).$ As above, ${\\mathcal{C}}_{o,>}(h)$\nis an open set, and therefore arcwise connected. Thus, for $z\\in I_g,$\nthere exists a continuous function $\\phi:[0,1]\\to {\\mathbb{R}}^2$ \nsuch that $\\phi(0)=o$ and $\\phi(1)=z.$ Since $\\phi$ is continuous,\nthe Im$(\\phi)$ is compact, and so \n$\\inf_{t\\in[0,1]}\\Psi(\\phi(t))$ is obtained, since $\\Psi$ is continuous\nby Proposition \\ref{prop:contfield}. Furthermore, since \nIm$(\\phi) \\subset {\\mathcal{C}}_{o,>}(h),$ we conclude that \n$\\inf_{t\\in[0,1]}\\Psi(\\phi(t))>h.$ Therefore, for some \n$h<g'<h_c$ we also have that $\\inf_{t\\in[0,1]}\\Psi(\\phi(t))>g'$,\nand so ${\\mathcal{C}}_{o,>}(g')$ contains an unbounded component.\nWe conclude that \n\\begin{equation} \\label{eqn:someg}\n{\\mathbb{P}}({\\mathcal{C}}_{o,>}(g) \\textrm{ is unbounded for some } g>h)\n={\\mathbb{P}}({\\mathcal{C}}_{o,>}(h) \\textrm{ is unbounded}).\n\\end{equation}\nCombining equations \\eqref{eqn:downlimg2} and \\eqref{eqn:someg} we conclude\nthat \\eqref{eqn:downlimg1} holds.\n\nIn order to complete the proof, we simply observe that for any \n$g<h$ we have that $\\theta_>(h)\\leq \\theta_{\\geq}(h)\\leq \\theta_>(g)$\nso that \n\\[\n\\theta_>(h)\\leq \\theta_{\\geq}(h) \n\\leq \\liminf_{g \\uparrow h} \\theta_{>}(g)=\\theta_>(h),\n\\]\nso that indeed $\\theta_>(h)=\\theta_{\\geq}(h)$ for every $h<h_c.$\n\\fbox{}\\\\\n\n\\noindent\n{\\bf Remark:}\nConsider the event $\\{o\\leftrightarrow \\partial B_n\\}$, that the \norigin is connected to the boundary of $B_n.$\nIn the discrete case, it is trivial that \n${\\mathbb{P}}_p(o \\leftrightarrow \\partial B_n)$ is continuous as a function of the percolation\nparameter $p,$  since it is an event that depends on the state of only finitely many\nedges. This then gives an easy proof of right-continuity (corresponding to left-continuity \nin our case). In our model, points of $\\eta$ at any distance contribute to \nthe field in $B_n.$ Therefore, we cannot claim immediate continuity of \n${\\mathbb{P}}(o \\leftrightarrow \\partial B_n)$, although our methods above can be used to prove\nit.\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{General aspects of baryogenesis}\n\nThe cosmological baryon asymmetry, the ratio of the baryon density to the\nentropy density of the universe,\n\\begin{equation}\nY_B = {n_B\\over s} = (0.6 - 1)\\cdot 10^{-10}\\;,\n\\end{equation}\ncan be understood as a consequence of baryon number violation, C and\nCP violation, and a deviation from thermal equilibrium\\cite{sac}. The\npresently observed value of the baryon asymmetry is then explained as a \nconsequence of the spectrum and interactions of elementary particles, \ntogether with the cosmological evolution.\n\nA crucial ingredient of baryogenesis is the connection between baryon number\n($B$) and lepton number ($L$) in the high-temperature, symmetric phase of\nthe standard model. Due to the chiral nature of the weak interactions $B$ and\n$L$ are not conserved\\cite{thoo}. At zero \ntemperature this has no observable effect due to the smallness of the weak \ncoupling. However, as the temperature approaches the critical temperature \n$T_c$ of the electroweak phase transition, $B$ and $L$ violating processes \ncome into thermal equilibrium\\cite{krs}. The rate of these processes is\nrelated to the free energy of sphaleron-type field configurations which carry\ntopological charge. In the standard model they lead to an effective\ninteraction of all left-handed fermions (cf. fig.~\\ref{fig_sphal}) which \nviolates baryon and lepton number by three units, \n\\begin{equation} \n    \\Delta B = \\Delta L = 3\\;. \\label{sphal1}\n\\end{equation}\nThe evaluation of the sphaleron rate in the symmetric high temperature phase is\na challenging problem\\cite{arn}. Although a complete theoretical understanding\nhas not yet been achieved, it is generally believed that $B$ and $L$ violating\nprocesses are in thermal equilibrium for temperatures in the range\n\\begin{equation} \nT_c^{EW} \\sim 100\\ \\mbox{GeV} < T < T_{SPH} \\sim 10^{12}\\ \\mbox{GeV}\\;.\n\\end{equation}\n\n  \\begin{figure}[t]\n    \\begin{center}\n      \\scaleboxto(7,7){\n        \\parbox[c]{9cm}{\\input{Fig01.tex}}\n      }\n    \\end{center}\n    \\caption{\\it One of the 12-fermion processes which are in thermal \n      equilibrium in the high-temperature phase of the standard model.\n      \\label{fig_sphal}\n    }\n  \\end{figure}\n  \nSphaleron processes have a profound effect on the generation of the\ncosmological baryon asymmetry.  Eq.~\\ref{sphal1} suggests that any\n$B+L$ asymmetry generated before the electroweak phase transition,\ni.e., at temperatures $T>T_c^{EW}$, will be washed out. However, since\nonly left-handed fields couple to sphalerons, a non-zero value of\n$B+L$ can persist in the high-temperature, symmetric phase if there\nexists a non-vanishing $B-L$ asymmetry. An analysis of the chemical potentials\nof all particle species in the high-temperature phase yields the following\nrelation between the baryon asymmetry\n$Y_B = (n_B-n_{\\overline{B}})\/s$ and the corresponding\n$L$ and $B-L$ asymmetries $Y_L$ and $Y_{B-L}$, respectively\\cite{chem},\n\\begin{equation}\\label{basic}\nY_B\\ =\\ C\\ Y_{B-L}\\ =\\ {C\\over C-1}\\ Y_L\\;.\n\\end{equation}\nHere $C$ is a number ${\\cal O}(1)$. In the standard model with three \ngenerations and one Higgs doublet one has $C=28\/79$. \n  \nWe conclude that $B-L$ violation is needed if the baryon asymmetry is\ngenerated before the electroweak transition, i.e. at temperatures \n$T > T_c^{EW} \\sim 100$~GeV.\nIn the standard model, as well as its supersymmetric version and its unified \nextensions based on the gauge group SU(5), $B-L$ is a conserved quantity. \nHence, no baryon asymmetry can be generated dynamically in these models.\n\n  \\begin{figure}[b]\n     \\input{Fig03.tex}\n     \\caption{\\it Lepton number violating lepton Higgs scattering\n       \\label{lept_fig}}\n  \\end{figure}\n\nThe remnant of lepton number violation in extensions of the standard model\nat low energies is the appearance of an effective $\\Delta L=2$ interaction \nbetween lepton and Higgs fields,\n  \\begin{equation}\\label{dl2}\n  {\\cal L}_{\\Delta L=2} = {1\\over 2}\\overline{l_L}\\,\\phi\\,g_{\\nu}\\,{1\\over M}\\,\n         g_{\\nu}^T\\,\\phi\\,l_L^c +\\mbox{ h.c.}\\;.\\label{intl2}\n  \\end{equation}\nSuch an interaction arises in particular from the exchange of heavy Majorana\nneutrinos (cf.~fig.~\\ref{lept_fig}). In the Higgs phase of the standard\nmodel, where the Higgs field aquires a vacuum expectation value, it gives\nrise to Majorana masses of the light neutrinos $\\nu_e$, $\\nu_\\mu$ and $\\nu_\\tau$.   \n\nAt finite temperature the $\\Delta L=2$ processes described by (\\ref{dl2}) take\nplace with the rate\\cite{fy1}\n  \\begin{equation}\n    \\Gamma_{\\Delta L=2} (T) = {1\\over \\pi^3}\\,{T^3\\over v^4}\\, \n    \\sum_{i=e,\\mu,\\tau} m_{\\nu_i}^2\\; .\n  \\end{equation}\nIn thermal equilibrium this yields an additional relation between the\nchemical potentials which implies\n\\begin{equation}\nY_B\\ =\\ Y_{B-L}\\ =\\ Y_L\\ =\\ 0 \\; .\n\\end{equation}\n\nTo avoid this conclusion, the $\\Delta L=2$ interaction (\\ref{intl2}) must not \nreach thermal equilibrium. For baryogenesis at very high temperatures, \n$T > T_{SPH} \\sim 10^{12}$ GeV, one has to require \n$\\Gamma_{\\Delta L=2} < H|_{T_{SPH}}$,\nwhere $H$ is the Hubble parameter. This yields a stringent upper bound on\nMajorana neutrino masses,\n\\begin{equation}\\label{nbound}\n\\sum_{i=e,\\mu,\\tau} m_{\\nu_i}^2 < (0.2\\ \\mbox{eV})^2\\;.\n\\end{equation}\nThis bound is comparable to the upper bound on the electron neutrino mass \nobtained from neutrinoless double beta decay. Note, however, that the bound\nalso applies to the $\\tau$-neutrino mass. In supersymmetric\ntheories two chiral U(1) symmetries in addition to baryon and lepton number\nare approximately conserved at temperatures above \n$T_{SS} \\sim 10^7$ GeV\\cite{iba}. This relaxes the upper bound (\\ref{nbound})\nfrom 0.2~eV to about 60~eV.\n  \nThe connection between lepton number and the baryon asymmetry is lost\nif baryogenesis takes place at or below the Fermi scale\\cite{dol}. However, \ndetailed studies of the thermodynamics of the electroweak transition have\nshown that, at least in the standard model, the deviation from thermal\nequilibrium is not sufficient for baryogenesis\\cite{jansen}. In the minimal \nsupersymmetric extension of the standard model (MSSM) such a scenario \nappears still possible for a limited range of parameters\\cite{dol}.\n\n\\section{Decays of heavy Majorana neutrinos}\n\nBaryogenesis above the Fermi scale requires $B-L$ violation, and therefore \n$L$ violation. Lepton number violation is most simply realized by adding \nright-handed Majorana neutrinos to the standard model.  Heavy right-handed \nMajorana neutrinos, whose existence is predicted by theories based on gauge \ngroups containing the Pati-Salam symmetry\\cite{pat} \nSU(4)$\\otimes$SU(2)$_L\\otimes$SU(2)$_R$, can also explain the smallness of \nthe light neutrino masses via the see-saw mechanism\\cite{seesaw}.\n\nThe most general lagrangian for couplings and masses of charged\nleptons and neutrinos reads \n  \\begin{equation}\\label{yuk}\n    {\\cal L}_Y = -\\overline{l_L}\\,\\wt{\\phi}\\,g_l\\,e_R\n            -\\overline{l_L}\\,\\phi\\,g_{\\nu}\\,\\nu_R\n            -{1\\over2}\\,\\overline{\\nu^C_R}\\,M\\,\\nu_R\n            +\\mbox{ h.c.}\\;.\n  \\end{equation}\nThe vacuum expectation value of the Higgs field $\\VEV{\\varphi^0}=v\\ne0$\ngenerates Dirac masses $m_l$ and $m_D$ for charged leptons and neutrinos,\n$m_l=g_lv$ and $m_D=g_{\\nu}v$, respectively, which are assumed to be much \nsmaller than the Majorana masses $M$.\nThis yields light and heavy neutrino mass eigenstates\n  \\begin{equation}\n     \\nu\\simeq K^{\\dg}\\nu_L+\\nu_L^C K\\quad,\\qquad\n     N\\simeq\\nu_R+\\nu_R^C\\, ,\n  \\end{equation}\nwith masses\n  \\begin{equation}\n     m_{\\nu}\\simeq- K^{\\dg}m_D{1\\over M}m_D^T K^*\\,\n     \\quad,\\quad  m_N\\simeq M\\, .\n     \\label{seesaw}\n  \\end{equation}\n  Here $K$ is a unitary matrix which relates weak and mass eigenstates. \n  \n  The right-handed neutrinos, whose exchange may erase any lepton\n  asymmetry, can also generate a lepton asymmetry by means of\n  out-of-equilibrium decays. This lepton asymmetry is then partially \n  transformed into a baryon asymmetry by sphaleron processes\\cite{fy}.  \n  The decay width of the heavy neutrino $N_i$ reads at tree level,\n  \\begin{eqnarray}\n    \\Gamma_{Di}&=&\\Gamma\\left(N^i\\to\\phi^c+l\\right)+\\Gamma\\left(N^i\\to\\phi+l^c\\right)\\nonumber\\\\\n           &=&{1\\over8\\pi}{(m_D^{\\dg}m_D)_{ii}\\over v^2}M_i\\;.\n    \\label{decay}\n  \\end{eqnarray}\nFrom the decay width one obtains an upper bound on the light neutrino masses\n  via the out-of-equilibrium condition\\cite{fisch}.\n  Requiring $\\Gamma_{D1}< H|_{T=M_1}$ yields the constraint\n  \\begin{equation}\\label{ooeb}\n  \\wt{m}_1\\ =\\ {{(m_D^{\\dg}m_D)_{11}}\\over M_1}\\ < \\ 10^{-3}\\, \\mbox{eV}\\;.\n  \\end{equation}\nMore direct bounds on the light neutrino masses depend on the structure\nof the Dirac neutrino mass matrix as we shall discuss below.\n\nInterference between the tree-level amplitude and the one-loop \nself-energy and vertex corrections yields the $CP$ asymmetry\\cite{cov,bp2}\n\\begin{eqnarray}\n\\varepsilon_1&=&{\\Gamma(N_1\\rightarrow l \\phi^c)-\\Gamma(N_1\\rightarrow l^c \\phi)\\over\n         \\Gamma(N_1\\rightarrow l \\phi^c)+\\Gamma(N_1\\rightarrow l^c \\phi)}\\nonumber\\\\\n     &\\simeq&{3\\over16\\pi v^2}\\;{1\\over\\left(m_D^{\\dag}m_D\\right)_{11}}\n      \\sum_{i=2,3}\\mbox{Im}\\left[\\left(m_D^{\\dag}m_D\\right)_{1i}^2\\right]\n      {M_1\\over M_i}\\label{cpa}\\;.\n\\end{eqnarray}\nHere we have assumed $M_1\\ll M_2,M_3$. For very small mass differences,\nwhich are comparable to the decay widths, one obtains a resonance \nenhancement\\cite{pil}. \n\nThe CP asymmetry (\\ref{cpa}) leads to the generated lepton \nasymmetry\\cite{kw},\n\\begin{equation}\nY_L\\ =\\ {n_L-n_{\\overline{L}}\\over s}\\ =\\ \\kappa\\ {\\varepsilon_1\\over g_*}\\;.\n    \\label{esti}\n\\end{equation}\nHere the factor $\\kappa<1$ represents the effect of washout processes. In order\nto determine $\\kappa$ one has to solve the full Boltzmann equations. In the\nexamples discussed below one has $\\kappa\\simeq 0.1\\ldots 0.01$.\n\nThe CP asymmetry (\\ref{cpa}) is given in terms of the Dirac and the Majorana\nneutrino mass matrices. One can always choose a basis for the right-handed\nneutrinos where the Majorana mass $M$ is diagonal with real and positive \neigenvalues. $m_D$ is a general complex matrix, which can be diagonalized by \na biunitary transformation. One then has\n\\begin{equation}\n    m_D=V\\,\\left(\\begin{array}{ccc}m_1&0&0\\\\0&m_2&0\\\\0&0&m_3\n              \\end{array}\\right)\\,U^{\\dag} \\;,\\quad\n    M=\\left(\\begin{array}{ccc}M_1&0&0\\\\0&M_2&0\\\\0&0&M_3\n    \\end{array}\\right)\\;,\n\\end{equation}\n  where $V$ and $U$ are unitary matrices and the $m_i$ are real and\n  positive. In the absence of a Majorana mass term $V$ and $U$ would \n  correspond to Kobayashi-Maskawa type mixing matrices of left- and \n  right-handed charged currents, respectively.\n\n \\begin{figure}[t]\n    \\mbox{ }\\hfill\n    \\epsfig{file=Fig06a.eps,width=8.2cm}\n    \\hfill\\mbox{ }\n    \\caption{\\it Time evolution of the neutrino number density and the\n     B-L asymmetry for $\\lambda\\simeq 0.1$ and $m_3\\simeq m_t$\\label{asyNS}}\n  \\end{figure} \n  \n  Note, that according to eqs.~(\\ref{decay}) and (\\ref{cpa}) the $CP$\n  asymmetry is determined by the mixings and phases present in the\n  product $m_D^{\\dg}m_D$, where the matrix $V$ drops out.  Hence, to\n  leading order, the mixings and phases which are responsible for\n  baryogenesis are entirely determined by the matrix $U$.\n  Correspondingly, the mixing matrix $K$ in the leptonic charged\n  current, which determines $CP$ violation and mixings of the light\n  leptons, depends on mass ratios and mixing angles and phases of $U$\n  and $V$.  This implies that there exists no direct connection\n  between $CP$ violation and generation mixing which are relevant at high\n  energies and at low energies, respectively.\n \nIn many models the quark and lepton mass hierarchies and mixings are \nparametrised in terms of a common mixing parameter $\\lambda \\sim 0.1$. Assuming\na hierarchy for the right-handed neutrino masses similar to the one\nsatisfied by up-type quarks,\n\\begin{equation}\n{M_1\\over M_2}\\ \\sim\\ {M_2\\over M_3}\\ \\sim\\ \\lambda^2\\;,\n\\end{equation}\nand a corresponding CP asymmetry \n\\begin{equation}\n\\varepsilon_1\\ \\sim\\ {\\lambda^4\\over 16\\pi}\\ {m_3^2\\over v^2}\\ \n       \\sim\\ 10^{-6}\\ {m_3^2\\over v^2} \\;,\n\\end{equation} \none obtains indeed the correct order of magnitude for the baryon \nasymmetry\\cite{bp} if one chooses $m_3\\simeq m_t\\simeq 174$ GeV, as expected \nin theories with Pati-Salam symmetry. Using as a constraint the \nvalue for the $\\nu_{\\mu}$-mass which is preferred by the MSW \nexplanation\\cite{msw} of the solar neutrino deficit\\cite{tot}, \n$m_{\\nu_{\\mu}}\\simeq 3\\cdot10^{-3}$~eV, the ansatz\\cite{bp} implies for the \nother light and the heavy neutrino masses \n\\begin{equation}\nm_{\\nu_e}\\simeq 8\\cdot 10^{-6}\\ \\mbox{eV}\\;,\n\\quad m_{\\nu_{\\tau}}\\simeq 0.15\\ \\mbox{eV}\\;, \n\\qquad M_3 \\simeq 2\\cdot10^{14}\\ \\mbox{GeV}\\;. \\label{nmass} \n\\end{equation}\nConsequently, one has $M_1\\simeq 2\\cdot10^{10}$ GeV and\n$M_2\\simeq 2\\cdot10^{12}$ GeV. The solution of the Boltzmann equations \nthen yields the baryon asymmetry (see fig.~\\ref{asyNS}),\n  \\begin{equation}\n     Y_B \\simeq 9\\cdot10^{-11}\\; , \\label{nonsusy_res1}\n  \\end{equation}\nwhich is indeed the correct order of magnitude. The precise value\ndepends on unknown phases.\n\n  \\begin{figure}[t]\n     \\begin{center}\n     \\epsfig{file=Fig10a.eps,width=8.2cm}\n     \\end{center}\n     \\caption{\\it Time evolution of the neutrino and the scalar neutrino\n       number densities, and of the lepton asymmetries for $\\lambda\\simeq 0.1$\n       and $m_3\\simeq m_t$. \\label{asyS}}\n  \\end{figure}\n  \nThe large mass $M_3$ of the heavy Majorana neutrino $N_3$\n(cf.~(\\ref{nmass})), suggests that $B-L$ is already broken at the\nunification scale $\\Lambda_{\\mbox{\\scriptsize GUT}} \\sim 10^{16}$\nGeV, without any intermediate scale of symmetry breaking. This large\nvalue of $M_3$ is a consequence of the choice $m_3 \\simeq m_t$. This\nis indeed necessary in order to obtain  sufficiently large CP asymmetry.\n\nThe recently reported atmospheric neutrino anomaly\\cite{kamio} may be\ndue to $\\nu_\\mu$-$\\nu_\\tau$ oscillations. The required mass difference is\n$\\Delta m^2_{\\nu_\\mu \\nu_\\tau} \\simeq (5\\cdot 10^{-4}-6\\cdot 10^{-3})$ eV$^2$,\ntogether with a large mixing angle $\\sin^2{2\\Theta}>0.82$. In the case\nof hierarchical neutrinos this corresponds to a $\\tau$-neutrino mass \n$m_{\\nu_\\tau} \\sim (0.02-0.08)$ eV. Within the theoretical uncertainties\nthis is consistent with the $\\tau$-neutrino mass (\\ref{nmass}) obtained from \nbaryogenesis. The $\\nu_\\tau$-$\\nu_\\mu$ mixing angle is not constrained by \nleptogenesis and therefore a free parameter in principle. The large value,\nhowever, is different from the mixing angles known in the quark sector\nand requires an explanation. An possible framework are U(1) family\nsymmetries\\cite{sat}. A large mixing angle can also naturally occur \ntogether with a mass hierarchy of light and heavy Majorana \nneutrinos\\cite{kug,buy}.\n\nWithout an intermediate scale of symmetry breaking, the unification\nof gauge couplings appears to require low-energy supersymmetry.\nSupersymmetric leptogenesis\\cite{camp} has recently been studied in detail,\ntaking all relevant scattering processes into account, which is necessary \nin order to get a reliable relation between the input parameters and the \nfinal baryon asymmetry\\cite{plue}. It turns out that the lepton number \nviolating scatterings are qualitatively more important than in the \nnon-supersymmetric case and that they can also account for the generation of \nan equilibrium distribution of heavy neutrinos at high temperatures.\n   \nThe supersymmetric generalization of the lagrangian (\\ref{yuk}) is\nthe superpotential\n\\begin{equation}\n    W = {1\\over2}N^cMN^c + \\mu H_1\\epsilon H_2 + H_1 \\epsilon L \\lambda_l E^c \n        + H_2 \\epsilon L \\lambda_{\\nu} N^c\\;,\n\\end{equation}\nwhere, in the usual notation, $H_1$, $H_2$, $L$, $E^c$ and $N^c$ are\nchiral superfields describing spin-0 and spin-${1\\over 2}$ fields.\nThe vacuum expectation value $v_2=\\left\\langle H_2\\right\\rangle$ of the \nHiggs field $H_2$ generates the Dirac mass matrix $m_D=\\lambda_{\\nu}v_2$ for \nthe neutrinos and their scalar partners.\n\nThe heavy neutrinos $N_i$ and their scalar partners $\\wt{N_i}$ decay with\ndifferent probabilities into final states with different lepton number. \nThe generated lepton asymmetries are shown in fig.~\\ref{asyS}\\cite{plue}. \n$Y_{L_f}$ and $Y_{L_s}$ denote the absolute values of the asymmetries stored \nin leptons and their scalar partners, respectively. They are related\nto the baryon asymmetry by\n\\begin{equation}\n    Y_B = - {8\\over 23}\\ Y_L\\quad, \\qquad Y_L = Y_{L_f} + Y_{L_s}\\;.\n\\end{equation} \n$Y_{N_1}$ is the number of heavy neutrinos per comoving volume\nelement, and\n\\begin{equation}\n    Y_{1\\pm} = Y_{\\widetilde{N_1^c}}\\pm Y_{\\wt{N}_1},\n\\end{equation}\nwhere $Y_{\\widetilde{N_1^c}}$ is the number of scalar neutrinos per comoving\nvolume element. As fig.~\\ref{asyS} shows, the generated baryon\nasymmetry has the correct order of magnitude, as in the non-supersymmetric \ncase.\n\nFrom the discussion of the out-of-equilibrium condition we know that the \ngenerated baryon asymmetry is very sensitive to the decay width $\\Gamma_{D1}$ \nof $N_1$, and therfore to ${(m_D^{\\dg}m_D)_{11}}$. In fact, the asymmetry essentially depends \non the effective neutrino mass $\\wt{m}_1$\\cite{plue}. For the case of \nhierarchical neutrino masses described above, one has\\cite{bp}\n\\begin{equation}\n\\wt{m}_1\\ =\\ {{(m_D^{\\dg}m_D)_{11}}\\over M_1}\\ \\simeq\\ m_{\\nu_\\mu}\\;.\n\\end{equation}\nIt turns out that a sufficiently large baryon asymmetry is generated in the\nrange\\cite{plue}\n\\begin{equation}\n    10^{-5}\\;\\mbox{eV}\\;\\ltap\\;\\wt{m}_1\\;\\ltap\\;\n    5\\cdot10^{-3}\\;\\mbox{eV}\\;,\n\\end{equation}\nwhich is consistent with the rough bound (\\ref{ooeb}).\nThis result is independent of any assumptions on the mass matrices. It is \nvery interesting that the $\\nu_{\\mu}$-mass preferred by the MSW explanation \nof the solar neutrino deficit lies indeed in the interval allowed by \nbaryogenesis!\n\nComparing non-supersymmetric and supersymmetric leptogenesis one sees \nthat the larger $CP$ asymmetry and the additional contributions from the \nsneutrino decays in the supersymmetric scenario are compensated by the \nwash-out processes which are stronger than in the non-supersymmetric case. \nThe final asymmetries are the same in the non-supersymmetric and in the \nsupersymmetric case.\n\nLeptogenesis can also be considered in extended models which contain\nheavy SU(2)-triplet Higgs fields in addition to right-handed \nneutrinos\\cite{ma,laz}. Decays of the heavy scalar bosons can in principle\nalso contribute to the baryon asymmetry. However, since these Higgs particles\ncarry gauge quantum numbers they are strongly coupled to the plasma and\nit is difficult to satisfy the out-of-equilibrium condition. The resulting\nlarge baryogenesis temperature is in conflict with the \n`gravitino constraint'\\cite{del}.\n\n\\section{SUSY mass spectrum and dark matter}\n\nThe out-of-equilibrium condition for the decay of the heavy Majorana\nneutrinos, the see-saw mechanism and the experimental evidence for small\nneutrino masses are all consistent and suggest rather large heavy neutrino\nmasses and a correspondingly large baryogenesis temperature. Within the\nansatz described in the previous section one obtains\n\\begin{equation}\nT_B\\ \\sim\\ M_1\\sim\\ 10^{10}\\ \\mbox{GeV}\\;.\n\\end{equation}\nSuch a large baryogenesis temperature can only be avoided in the very\nspecial case of a strong resonant amplification of the CP violating \ndecays\\cite{pil}. \n\nIn the particularly attractive supersymmetric version of leptogenesis one \nalso has to consider the following two issues: the size of other\npossible contributions to the baryon asymmetry and the consistency of\nthe large baryogenesis temprature with the `gravitino constraint'\\cite{khl}.\nA large asymmetry may potentially be generated by coherent oscillations\nof scalar fields which carry baryon and lepton number\\cite{din}. However,\nit appears likely that\nthe interactions of the right-handed neutrinos are sufficient to erase\nsuch large primordial baryon and lepton asymmetries\\cite{jak}. \n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfig{file=Gluon_t.ps}\n\\end{center}\n\\caption{\\it Typical gravitino production processes mediated by gluon \nexchange. \\label{gravprod}}\n\\end{figure}\n\nThe `gravitino contraint' is particularly interesting since it is model \nindependent to a large extent, and it therefore provides very interesting\ninformation about possible extensions of the standard model.\nThe production of gravitinos ($\\tilde{G}$) at high temperatures is dominated\nby two-body processes involving gluinos ($\\gl$) (cf.~fig.~\\ref{gravprod}). \nOn dimensional grounds the production rate has the form\n\\begin{equation}\n\\Gamma(T)\\ \\propto\\ {1\\over M^2}\\ T^3\\;,\n\\end{equation}\nwhere $M = (8\\pi G_N)^{-1\/2} = 2.4\\cdot 10^{18}$ GeV is the Planck mass. \nHence, the density of thermally produced gravitinos increases strongly with \ntemperature.\n\nThe production cross section is enhanced by a factor \n$(m_{\\gl}\/m_{\\tilde{G}})^2$ for light gravitinos\\cite{mor}.\nThe thermally averaged cross section has recently been evaluated for arbitrary\ngravitino masses. The result reads\\cite{bol}\n\\begin{eqnarray}\nC(T) &=& \\VEV{\\sigma_{(L)} v\\sbs{rel}} \\nonumber\\\\\n&=& {21 g^2(T)\\over 32\\pi\\zeta^2(3) M^2}((N^2-1)C_A + 2n_f N C_F)\n\\left(1+{m_{\\gl}^2(T) \\over 3 m_{\\tilde{G}}^2}\\right)\\nonumber\\\\\n&&\\hspace{2cm}\\left(\\ln{1\\over g^2(T)} + {5\\over 2} + 2\\ln{2} -2\\gamma_E\\right)\\;.\n\\end{eqnarray}\nHere $C_A$ and $C_F$ are the usual colour factors for the group SU(N) and \n$2n_f$ is the number of colour-triplet chiral multiplets, i.e. $2n_f=12$ in \nthe MSSM. The logarithmic collinear singularity of the cross section has\nbeen regularized by a plasma mass $m \\sim g(T)T$ for the gluon. \nThe unknown constant part of the thermally averaged cross section is expected\nto be of the same size as the term proportional to $\\ln(1\/g^2(T))$.\nFor QCD (N=3) one has\n\\begin{equation}\\label{eq:2}\nC(T) \\simeq 10 {g^2(T)\\over M^2}\\left(1+{m_{\\gl}^2(T) \\over 3 m_{\\tilde{G}}^2}\\right)\n\\left(\\ln{1\\over g^2(T)} + 2.7\\right)\\;.\n\\end{equation}.\n\nThe cross section $C(T)$ enters in the Boltzmann equation, which \ndescribes the generation of a gravitino density $n_{\\tilde{G}}$ in the thermal \nbath\\cite{kol},\n  \\begin{equation}\n    \\frac{dn_{\\tilde{G}}}{dt} + 3 H n_{\\tilde{G}} = C(T) n\\sbs{rad}^2\\;.\n    \\label{eq:3}\n  \\end{equation}\nHere $H(T)$ is the Hubble parameter and $n\\sbs{rad}=\\frac{\\zeta(3)}{\\pi^2}T^3$ \nis the number density of a relativistic bosonic degree of freedom. \nFrom eqs.~(\\ref{eq:2}) and (\\ref{eq:3}) one obtains for the density of\nlight gravitinos and the corresponding contribution to $\\Omega h^2$ at\ntemperatures $T<1$~MeV, i.e. after nucleosynthesis, \n  \\begin{equation}\n    Y_{\\tilde{G}}\\simeq 3.2\\cdot 10^{-10}\n    \\left(\\frac{T_B}{10^{10}\\,\\mbox{GeV}}\\right)\n    \\left(\\frac{100\\,\\mbox{GeV}}{m_{\\tilde{G}}}\\right)^2\n    \\left(\\frac{m_{\\gl}(\\mu)}{1\\,\\mbox{TeV}}\\right)^2,\n    \\label{eq:6}\n  \\end{equation}\n\n  \\begin{eqnarray}\n    \\Omega_{\\tilde{G}}h^2 & = & m_{\\tilde{G}} Y_{\\tilde{G}}(T) n\\sbs{rad}(T) \\rho_c^{-1} \\nonumber \\\\\n    & \\simeq & 0.60\n    \\left(\\frac{T_B}{10^{10}\\,\\mbox{GeV}}\\right)\n    \\left(\\frac{100\\,\\mbox{GeV}}{m_{\\tilde{G}}}\\right)\n    \\left(\\frac{m_{\\gl}(\\mu)}{1\\,\\mbox{TeV}}\\right)^2.    \n    \\label{eq:7}\n  \\end{eqnarray}\nHere we have used $g(T_B)=0.85$; $\\rho_c=3H_0^2M^2$ is the critical energy\ndensity, and $m_{\\gl}(T)=\\frac{g^2(T)}{g^2(\\mu)} m_{\\gl}(\\mu)\\gg m_{\\tilde{G}}$,\nwith $\\mu \\sim 100$GeV. \n\nThe primordial synthesis of light elements (BBN) yields stringent\nconstraints on the amount of energy which may be released after\nnucleosynthesis by the decay of heavy nonrelativistic particles into\nelectromagnetically  and strongly interacting relativistic particles. These \nconstraints have been studied in detail by several groups\\cite{ell,kaw,holt}. \nDepending on the lifetime of the decaying particle $X$ its\nenergy density cannot exceed an upper bound. Sufficient conditions\\cite{ell}\nare\n  \\begin{eqnarray}\n    \\mbox{(I)} & m_X Y_X(T) < 4\\cdot 10^{-10}\\,\\mbox{GeV}, &\n    \\tau < 2\\cdot 10^6\\,\\mbox{sec},\n    \\label{eq:8} \\\\\n    \\mbox{(II)} & m_X Y_X(T) < 4\\cdot 10^{-12}\\,\\mbox{GeV}, &\n    \\tau\\,\\,\\mbox{arbitrary},\n    \\label{eq:9}\n  \\end{eqnarray}\nwhere $Y_X(T) = n_X(T)\/n\\sbs{rad}(T)$.\n\n\\begin{figure}[tb]\n\\begin{center}\n    \\vskip .1truein\n    \\centerline{\\epsfysize=10cm {\\epsffile{oh2co-mx.eps}}}\n    \\vskip -0.1truein\n    \\caption{\\it Neutralino relic density versus neutralino mass. The\n     horizontal lines bound the region $0.025<\\Omega_{\\chi}h^2<1$.}\\label{vary}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\n\nGravitinos interact only gravitationally. Hence, their existence leads\nalmost unavoidably to a density of heavy particles which decay after\nnucleosynthesis. The partial width for the decay of an unstable\ngravitino into a gauge boson $B$ and a bino $\\bi$ is given by\n($m_{\\bi}\\ll m_{\\tilde{G}}$)\\cite{kaw},\n  \\begin{equation}\n    \\Gamma(\\tilde{G} \\rightarrow B\\bi)  \\simeq  \n    \\frac{1}{32\\pi}\\frac{m_{\\tilde{G}}^3}{M^2} \n    \\simeq  \\left[ 4\\cdot10^8\\left( \\frac{100\\,\\mbox{GeV}}{m_{\\tilde{G}}}\n    \\right)^3\\,\\mbox{sec}\\right]^{-1}.\n    \\label{eq:10}\n  \\end{equation}\nIf for a fermion $\\psi$ the decay into a final state with a scalar $\\phi$ in\nthe same chiral multiplet and a gravitino is kinematically allowed, \nthe partial width reads ($m_{\\psi}\\gg m_{\\phi}$),\n\\begin{equation}\n    \\Gamma(\\psi \\rightarrow \\tilde{G} \\phi) =\\Gamma(\\psi \\rightarrow \\tilde{G} \\phi^*)\n     \\simeq \\frac{1}{96\\pi}\\frac{m_{\\psi}^5}{m_{\\tilde{G}}^2M^2}\\;. \n    \\label{eq:11}\n  \\end{equation}\nGiven these lifetimes and the mass spectrum of superparticles in the\nMSSM one can examine whether one of the conditions (I) and (II)\non the energy density after nucleosynthesis is satisfied.\n\nConsider first a typical example of supersymmetry breaking masses in\nthe MSSM, $m_{\\bi}<m_{\\tilde{G}}\\simeq 100\\,\\mbox{GeV}<m_{\\gl}\\simeq 500\\,\n\\mbox{GeV}$, and $T_B \\simeq 10^{10}$~GeV.\nFrom eqs.~(\\ref{eq:6}) and (\\ref{eq:10}) we conclude $\\tau_{\\tilde{G}}\\simeq\n4\\cdot10^8$~sec, $m_{\\tilde{G}}Y_{\\tilde{G}}(T)\\simeq 4\\cdot10^{-9}$~GeV. \nAccording to condition (II) (\\ref{eq:9}) this energy density\nexceeds the allowed maximal energy density by 3 orders of\nmagnitude. This clearly illustrates the `gravitino problem'!\nThe stringent constraints from BBN can be evaded for very light \ngravitinos\\cite{pag}, with $m_{\\tilde{G}}< 1$ keV and possibly also for very heavy\ngravitinos, with $m_{\\tilde{G}} = {\\cal O}(10$~TeV). \n\n\\begin{figure}[tb]\n\\begin{center}\n    \\vskip .1truein\n    \\centerline{\\epsfysize=10cm {\\epsffile{zgzh-mxco.eps}}}\n    \\vskip -0.1truein\n    \\caption{\\it Neutralino composition $Z_g\/(1-Z_g)$ versus neutralino mass\n     for $0.025<\\Omega_{\\chi}h^2<1$.}\\label{hole}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\n\nAnother interesting possiblity is the case where the gravitino is the LSP\nwith a mass ${\\cal O}(100$~GeV)\\cite{bol}. In this case\none has to worry about the decays of the next-to-lightest\nsuperparticle (NSP) after nucleosynthesis. The lifetime constraint of\ncondition (I), $\\tau\\sbs{NSP}<2\\cdot10^6$~sec, yields a lower bound on\nthe NSP mass which depends on the gravitino mass $m_{\\tilde{G}}$ \n(cf.~eq.~(\\ref{eq:11}) and fig.~\\ref{fig:1}).\n\nFor a large range  of parameters the NSP is a neutralino $\\chi$, \ni.e. a linear combination of higgsinos and gauginos,\n  \\begin{equation}\n    \\chi = N_1 \\bi + N_2 \\wi + N_3 \\hi + N_4 \\hj\\;. \n    \\label{eq:12}\n  \\end{equation}\nThe NSP density after nucleosynthesis has been studied in great detail\nby a number of authors \\cite{dre}, since the density of stable neutralinos\nwould contribute to dark matter. A systematic study for a large range of\nMSSM parameters has been carried out by Edsj\\\"o and Gondolo\\cite{eds}. \nVarying the MSSM parameters, $\\Omega_\\chi h^2$ varies over eight orders of \nmagnitude, from $10^{-4}$ to $10^4$ (see fig.~\\ref{vary}\\cite{eds}). Note, \nthat for a large part of parameter space one has $\\Omega_\\chi h^2 < 0.025$. In \nparticular this is the case for a higgsino-like neutralino, i.e. \n$Z_g=|N_1|^2+|N_2|^2<\\frac{1}{2}$, in the mass range\n$80\\,\\mbox{GeV}<m_\\chi <450\\,\\mbox{GeV}$\\cite{eds}. For these parameters \nneutralino pair annihilation into $W$ boson pairs is very efficient, which\nleads to the `higgsino hole' in fig.~\\ref{hole}\\cite{eds}.\n\n\\begin{figure}[tb]\n\\begin{center}\n   \\vskip .1truein\n    \\centerline{\\epsfysize=10cm {\\epsffile{fig1.eps}}}\n    \\vskip -0.3truein\n\\caption{\\it Upper and lower bounds on the NSP mass as function of the\ngravitino mass. The full lines represent the upper bound on the gluino \nmass $m_{\\gl}(\\mu) > m_{NSP}$ for different reheating \ntemperatures. The dashed line is the lower bound on $m_{NSP}$ which follows\nfrom the NSP lifetime. A higgsino-like NSP with a mass in the shaded area\nsatisfies all cosmological constraints including those from primordial\nnucleosynthesis.}\n    \\label{fig:1}\n\\end{center}  \n\\vspace{-0.5cm}\n\\end{figure}\n\nThe bound $\\Omega h^2<0.008$, which corresponds to the bound on the mass density \n$m_\\chi Y_\\chi (T)<4\\cdot10^{-10}$~GeV of condition (I),\nis satisfied for higgsino-like neutralinos\nin the mass range $80\\,\\mbox{GeV}<m_\\chi <300\\,\\mbox{GeV}$\\cite{gon}.\nWe conclude that higgsino-like NSPs in this mass range and with a lifetime\n$\\tau<2\\cdot10^6$~sec are compatible with the constraints from\nprimordial nucleosynthesis.\nNote that this is a sufficient, yet not necessary condition for satisfying\nthe bound $\\Omega h^2<0.008$. Very small neutralino densities are also obtained\nfor other sets of MSSM parameters.  \n  \\begin{figure}[tb]\n\\begin{center}\n    \\vskip .1truein\n    \\centerline{\\epsfysize=10cm {\\epsffile{fig2.eps}}}\n    \\vskip -0.1truein\n    \\caption{\\it Contribution of gravitinos to the density parameter $\\Omega h^2$\n     for different gravitino masses $m_{\\tilde{G}}$ as function of the reheating\n     temperature $T_B$. The gluino mass has been set to $m_{\\gl}(\\mu)=500$~GeV.}\n    \\label{fig:2}\n\\end{center}\n\\vspace{-0.5cm}\n  \\end{figure}\n\nFinally, we have to discuss the necessary condition that gravitinos do\nnot overclose the universe,\n  \\begin{equation}\n    \\Omega_{\\tilde{G}} h^2 < 1\\;.\n    \\label{eq:13}\n  \\end{equation}\nSince $m_{\\tilde{G}}<m_\\chi <m_{\\gl}$, the gravitino density is given by\neq.~(\\ref{eq:7}). Hence, the condition (\\ref{eq:13}) yields an upper bound on\nthe NSP mass $m_\\chi$ which depends on the gravitino mass and the baryogenesis \ntemperature. The different constraints are summarized in \nfig.~\\ref{fig:1}\\cite{bol} which illustrates that for a wide range of MSSM \nparameters, where\n  \\begin{equation}\n    m_{\\tilde{G}}<m_\\chi <m_{\\gl}\\;,  \\label{eq:14}\n  \\end{equation}\nand $80\\,\\mbox{GeV}<m_\\chi <300\\,\\mbox{GeV}$, the baryogenesis\ntemperature may be as large as ${\\cal O}(10^{10})$~GeV.\n\nIt is remarkable that for temperatures $T_B=10^8\\dots 10^{11}$~GeV,\nwhich are natural for leptogenesis, and for gravitino masses in the range\n$m_{\\tilde{G}}=10^0\\dots 10^3$~GeV, which is expected for gravity induced\nsupersymmetry breaking, the relic density of gravitinos is cosmologically\nimportant (cf.~fig.~\\ref{fig:2}).\nAs an example, for $T_B\\simeq10^{10}$~GeV, $m_{\\gl}(\\mu)\\simeq 500$~GeV, and\n$m_{\\tilde{G}}\\simeq 50$~GeV, one has $\\Omega_{\\tilde{G}} h^2 \\simeq 0.30$.\\\\\n\n\\section{Summary}\n\nDetailed studies of the thermodynamics of the electroweak interactions at\nhigh temperatures have shown that in the standard model and most of its\nextensions the electroweak transition is too weak to affect the \ncosmological baryon asymmetry. Hence, one has to search for baryogenesis\nmechanisms above the Fermi scale. \n\nDue to sphaleron processes baryon number and lepton number are related\nin the high-temperature, symmetric phase of the standard model. As a\nconsequence, the cosmological baryon asymmetry is related to neutrino\nproperties. Baryogenesis requires lepton number violation, which occurs\nin extensions of the standard model with right-handed neutrinos and\nMajorana neutrino masses. \n\nAlthough lepton number violation is needed in order to obtain a baryon\nasymmetry, it must not be too strong since otherwise any baryon and lepton\nasymmetry would be washed out. This leads to stringent upper bounds on\nneutrino masses which depend on the particle content of the theory.\n\nThe solar and atmospheric neutrino deficits can be interpreted as a result\nof neutrino oscillations. For hierarchical neutrinos the corresponding\nneutrino masses are very small. Assuming the see-saw mechanism, this suggests\nthe existence of very heavy right-handed neutrinos and a large scale of\n$B-L$ breaking.\n\nIt is remarkable that these hints on the nature of lepton number violation\nfit very well together with the idea of leptogenesis.\nFor hierarchical neutrino masses, with $B-L$ broken at the\nunification scale $\\Lambda_{\\mbox{\\scriptsize GUT}}\\sim 10^{16}\\;$GeV,  \nthe observed baryon asymmetry $n_B\/s \\sim 10^{-10}$ is naturally\nexplained by the decay of heavy Majorana neutrinos. The corresponding\nbaryogenesis temperature is $T_B \\sim 10^{10}$ GeV. \n\nIn supersymmetric models implications for the mass spectrum of superparticles \ncan be derived. The rather large baryogenesis temperature leads to a high\ndensity of gravitinos. Depending on the masses of the other superparticles \ntheir late decay may change the primordial abundances of light elements in\ndisagreement with observation. This `gravitino problem' can be avoided for\nvery light gravitinos, with $m_{\\tilde{G}}< 1$ keV, and possibly also for very\nheavy gravitinos, with $m_{\\tilde{G}} = {\\cal O}(10$~TeV). Another interesting\npossibility is that the gravitino is the LSP, with a higgsino as NSP.\nIt is intriguing that for a mass range $m_{\\tilde{G}}=10^0\\dots 10^2$~GeV and\nreheating temperatures $T_B=10^8\\dots 10^{11}$~GeV, which naturally occur\nafter inflation, one obtains a gravitino contribution to cold dark matter \nwith $\\Omega h^2 = {\\cal O}(1)$.   \n\n\\mbox{ }\\vspace{4ex}\\\\\n\\noindent\n\\setlength{\\parskip}{1ex}\n{\\bf Acknowledgments}\\\\\n\\mbox{ }\\\\    \nThe author would like to thank H.~J.~de~Vega and N.~S\\'anchez for\norganizing an enjoyable and stimulating colloquium.\n\\mbox{ }\\\\\\noindent\n\n\\clearpage\n\\mbox{ }\\vspace{4ex}\\\\\n\\noindent\n{\\bf\\Large References}\n\\noindent\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor many years the  Ly$\\alpha$~ forest has been\nconsidered a different class of objects with respect to galaxies.\nThe available \nsensitivity was too low to detect any sign of non--primordial\ncomposition in the intergalactic gas clouds at  high redshift.\nThanks to the advent of high resolution and signal--to--noise\nspectroscopy, the old idea on the majority of quasar absorption lines \nhas been \nrevisited and opened in the last few years a still pending debate\non the connection between the Ly$\\alpha$~ forest and the\ngalaxy formation of the early Universe. The detection of ions\ndifferent from CIV in optically thin\nLy$\\alpha$~ clouds \nis made complicated by harder\nobservational conditions, whereas the still too poor knowledge of the\nionisation mechanisms which determine the ion abundances in those\nclouds has often discouraged attempts of\nmetal content estimations as a function of\nredshift and of HI column density.\nHowever abundance investigation of the Ly$\\alpha$~ clouds has fundamental\nimplications in the understanding of the enrichment processes in the IGM\nby Pop III stars in the $z>5$ Universe.\n\n\\section{Metallicity measurements}\n\n\\begin{figure}\n\\centerline{\\vbox{\n\\psfig{figure=savaglioF1.ps,height=4cm,angle=0}\n}}\n\\caption[]{\\label{f0} Carbon abundance relative to solar as a function\nof redshift in the\noptically thin Ly$\\alpha$~ lines which both show CIV and\nSiIV doublets.}\n\\end{figure}\n\nThe sample of optically thin\nabsorption lines with  $14.5 < \\log N_{\\rm HI} < 16.5$ \nhas been obtained by high resolution spectroscopy, mainly HIRAS\/Keck\n(Songaila 1997b) but also by EMMI\/NTT for the $z\\gsim 3.7$ systems \n(Savaglio et al. 1997). For all the systems  CIV and\/or SiIV and CII\ndetections or upper limits are given in redshift coverage\n$2 < z < 4.5$. \nThe lower bound in $N_{\\rm HI}$ \nis due to the very rare metal detection in lower\ncolumn density systems. In this range even if the line can\nbe saturated (depending on the Doppler width)\n Monte Carlo simulations showed that fitting procedures\nof synthetic individual lines with similar resolution and S\/N ratio\nof the observed spectra give HI column density errors which are less\nthan a few tens of $dex$\n(for $b=25$ km~s$^{-1}$, $\\log N_{\\rm HI} = 15.5$, FWHM = 12 km~s$^{-1}$\nand S\/N = 20 this is typically 0.1 $dex$). The blending effect has a much\nmore dramatic impact on column density uncertainties\nand for this reason, we consider in the\ncase of complex structures as an individual cloud \nthe total column densities of HI and of\nmetal lines.\n\nEstimating the heavy element  content in the Ly$\\alpha$~ clouds is mostly\ncomplicated by the poor knowledge of the ionising sources. As a first\nsimplification, we assume that this is dominated by photoionisation\nof the UV background and neglect any other\nmechanism. \nCollisional ionisation is important when the gas temperature \nexceeds $10^5$ K. At that temperature, the\nDoppler parameter for HI is 41 km~s$^{-1}$, well above the mean value\ntypically found\nin Ly$\\alpha$~ clouds. The analysis of metal lines in Ly$\\alpha$~ clouds (Rauch\net al., 1997) shows that the mean ``Doppler'' temperature in these clouds is\n$\\sim4\\times10^4$ K, making any evidence of collisional ionisation\nhard to justify. \nOnce the photoionisation equilibrium is assumed, we first\nconsider the subsample of Ly$\\alpha$~ clouds which show both CIV and SiIV\nabsorption. To calculate the metallicity we use CLOUDY and assume six\ndifferent shapes for the UV background normalized to the value at\nthe Lyman limit ($J_{912} = 5\\times10^{-22}$ erg s$^{-1}$ cm$^{-2}$\nHz$^{-1}$ sr$^{-1}$) changing the parameter $S_L =\nJ_{912}\/J_{228}$ in the range $200-3000$. We varied \nthe [C\/H] and gas density in such a\nway to reproduce the observed CIV. We also assume the\nrelative silicon--to--carbon abundance to be between 0 and three times solar\nand consider the cloud size along the line of sight to be in the\nrange 1 kpc $\\lsim R \\lsim 50$ kpc. Given these assumptions, we\nobtain for this subsample a set of 18 [C\/H] measurements shown in\nFig.~\\ref{f0}.  Carbon abundance in clouds with detected\ncarbon and silicon  has a large spread with\nmean values of [C\/H] $= -1.8$ and no evidence of redshift evolution.  We\nnotice that this sample might consist of metal--rich Ly$\\alpha$~ clouds\nsince it has been selected because of the SiIV detection \nand might not be representative of the whole population of Ly$\\alpha$~ clouds. \nIn a recent work, Songaila (1997a) \nhas estimated the total universal metallicity\nat $z\\sim3$ (assuming that at that time the baryonic matter of the\nUniverse mostly resides in the Ly$\\alpha$~ forest)\nto be in the range 1\/2000 and 1\/630 relative to solar.\n\n\\begin{figure}\n\\centerline{\\vbox{\n\\psfig{figure=savaglioF2.ps,height=8.5cm,angle=0}\n}}\n\\caption[]{\\label{f1} Ion column density ratios as a function of\nredshift. Solid curves are models assuming the UVB \nas described by the first two panels ($\\log N_{\\rm HI} =\n15$ and [C\/H] $=-1.8$). Dashed curves are the same but for [C\/H] \n$=-1.8\\pm0.8$.}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\vbox{\n\\psfig{figure=savaglioF3.ps,height=4.9cm,angle=0}\n}}\n\\caption[]{\\label{f2} Ion column density ratios as a function of\nHI column density. Solid and dashed curves \nare model calculations assuming the UVB\nat $z=3$ of Fig.~\\ref{f1}  ([C\/H] $=-1.8\\pm0.8$). Straight lines\nrepresent detection limits.}\n\\end{figure}\n\nIn a different approach, we consider the whole sample\nand regard the global observed properties instead of the individual\n systems and compare with models. Results of column density ratios on the $z$ and\n$N_{\\rm HI}$ planes are shown in Figs.~\\ref{f1} and\n\\ref{f2}. In Fig.~2 we investigate the redshift evolution of\nobserved column densities in the case of $S_L$ and $J_{912}$ as reported.\nThe discussed trend of SiIV\/CIV (Cowie et al., this\nconference proceedings) can be\nreproduced by a redshift evolution of $S_L$ from 200 at $z\\sim2$ to 3000\nat $z\\sim4$. The same model can take into account other observed ion ratios.\nIn Fig.~3 we compare observations with CLOUDY\nmodels assuming that all the clouds of the sample are at the same mean\nredshift of $z=3$ with $S_L=800$ \nand the gas density proportional to the square root of\n$N_{\\rm HI}$, as given in the case of spherical clouds in\nphotoionisation equilibrium with the UVB. In both figures the solid\nlines are obtained for metallicity [C\/H] $= -1.8$ and [Si\/C] = [O\/C] =\n0.5, [N\/C] = 0. Models of photoionisation equilibrium \ncan include the majority of metal\ndetections (also considering the metallicity spread) but\nCII\/HI which, as function of $N_{\\rm HI}$, looks to be\nsteeper than calculated. Additional observations of CII would probably\ncast further light on the discussion on the ionisation state and metal content\nin the Ly$\\alpha$~ clouds. \nIn both figures, the numerous upper limits falling below the dashed curve\n[C\/H] $= -2.6$ is an indication that in many clouds the\nmetallicity is lower than the values found in the selected sample.\n\n\\section{The future}\n\nThe investigation of low and\nintermediate redshift ($z=1-2.5$) \nobservations of OVI and NV in $\\log N_{\\rm HI}\\lsim 14$\nLy$\\alpha$~ clouds might succeed in answering the question of how efficient the mixing\nprocesses\nin the IGM at high redshift has been. Relative abundances can\nprovide new hints on the study of metal production by Pop III\nstars. In particular NV since it has been predicted to be underproduced in\nmassive stars with low initial metallicity (Arnett 1995).\nMore observations of the SiIV\/CIV ratio for $z<2$\nand $z>4$ are a challenging probe of the redshift evolution of the UVB,\nthough this can be one of the many possible reasons for the\nobserved SiIV\/CIV trend (another would be redshift evolution\nof the gas density being lower at lower redshift). \nMore interesting conclusions await outcomes\nfrom new high quality data of Keck observations.\n\n\\begin{iapbib}{}{\n\n\\bibitem{}\nArnett D., 1995, ARA\\&A, 33, 115\n\\bibitem{}\nRauch M., Sargent W.L.W., Womble D.S., Barlow T.A., 1997, ApJ, 467, L5\n\\bibitem{}\nSavaglio S., Cristiani S., D'Odorico S., Fontana A., Giallongo E.,\nMolaro P., 1997, A\\&A, 318, 347\n\\bibitem{}\nSongaila A., 1997a, ApJL, {\\it in press}, astro--ph\/9709046\n\\bibitem{} \nSongaila A., 1997b, {\\it in preparation}\n}\n\\end{iapbib}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}