Journal of Statistical Physics

, Volume 154, Issue 6, pp 1461–1482 | Cite as

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

  • Hubert Lacoin


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 \({\mathbb Z} ^d\) with parameter \(p>p_c({\mathbb Z} ^d)\). The object of this paper is to study the connective constant of the dilute lattice \(\limsup _{N\rightarrow \infty } Z_N^{1/N}\), which is a non-random quantity. We want to investigate if the inequality \(\limsup _{N\rightarrow \infty } (Z_N)^{1/N} \le \lim _{N\rightarrow \infty } {\mathbb E} [Z_N]^{1/N}\) 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 \(d\) is sufficiently large there exists \(p^{(2)}_c>p_c\) such that the inequality is strict for \(p\in (p_c,p^{(2)}_c)\).


Percolation Self-avoiding walk Random media Polymers Disorder relevance 

Mathematics Subject Classification

82D60 60K37 82B44 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Place du Maréchal de Lattre de TassignyCEREMADEParis Cedex 16France

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