Existence of a Non-Averaging Regime for the Self-Avoiding Walk on a High-Dimensional Infinite Percolation Cluster

Let ZN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_N$$\end{document} be the number of self-avoiding paths of length N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document} starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb Z} ^d$$\end{document} with parameter p>pc(Zd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>p_c({\mathbb Z} ^d)$$\end{document}. The object of this paper is to study the connective constant of the dilute lattice lim supN→∞ZN1/N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup _{N\rightarrow \infty } Z_N^{1/N}$$\end{document}, which is a non-random quantity. We want to investigate if the inequality lim supN→∞(ZN)1/N≤limN→∞E[ZN]1/N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup _{N\rightarrow \infty } (Z_N)^{1/N} \le \lim _{N\rightarrow \infty } {\mathbb E} [Z_N]^{1/N}$$\end{document} obtained with the Borel–Cantelli Lemma is strict or not. In other words, we want to know if the quenched and annealed versions of the connective constant are equal. On a heuristic level, this indicates whether or not localization of the trajectories occurs. We prove that when d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document} is sufficiently large there exists pc(2)>pc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{(2)}_c>p_c$$\end{document} such that the inequality is strict for p∈(pc,pc(2))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (p_c,p^{(2)}_c)$$\end{document}.