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Let Z_N be the number of self-avoiding paths of length N starting from theorigin on the infinite cluster obtained after performing Bernoulli percolationon Z^d with parameter p>p_c(Z^d). The object of this paper is to study theconnective 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} obtainedwith the Borel-Cantelli Lemma is strict or not. In other words, we want to knowthe the quenched and annealed versions of the connective constant are the same.On a heuristic level, this indicates whether or not localization of thetrajectories occurs. We prove that when d is sufficiently large there existsp^{(2)}_c>p_c such that the inequality is strict for p\in (p_c,p^{(2)}_c).

Existence of a non-averaging regime for the self-avoiding walk on a high-dimensional infinite percolation cluster