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Article Dans Une Revue Journal of Statistical Physics Année : 2014

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

Résumé

Let Z_N be the number of self-avoiding paths of length N starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on Z^d with parameter p>p_c(Z^d). The object of this paper is to study the connective constant of the dilute lattice \limsup_{N\to \infty} Z_N^{1/N}, which is a non-random quantity. We want to investigate if the inequality \limsup_{N\to \infty} (Z_N)^{1/N} \le \lim_{N\to \infty} E[Z_N]^{1/N} obtained with the Borel-Cantelli Lemma is strict or not. In other words, we want to know the the quenched and annealed versions of the connective constant are the same. On a heuristic level, this indicates whether or not localization of the trajectories occurs. We prove that when d is sufficiently large there exists p^{(2)}_c>p_c such that the inequality is strict for p\in (p_c,p^{(2)}_c).

Dates et versions

hal-00922723 , version 1 (30-12-2013)

Identifiants

Citer

Hubert Lacoin. Existence of a Non-Averaging Regime for the Self-Avoiding Walk on a High-Dimensional Infinite Percolation Cluster. Journal of Statistical Physics, 2014, 154 (6), pp.1461-1482. ⟨10.1007/s10955-014-0926-x⟩. ⟨hal-00922723⟩
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