Abstract
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 \(p\in (p_c,p^{(2)}_c)\).
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Lacoin, H. Existence of a Non-Averaging Regime for the Self-Avoiding Walk on a High-Dimensional Infinite Percolation Cluster. J Stat Phys 154, 1461–1482 (2014). https://doi.org/10.1007/s10955-014-0926-x
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DOI: https://doi.org/10.1007/s10955-014-0926-x