Non-coincidence of Quenched and Annealed Connective Constants on the supercritical planar percolation cluster
Résumé
In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on $\bbZ^d$. More precisely, we count ZN the number of self-avoiding paths of length N on the infinite cluster, starting from the origin (that we condition to be in the cluster). We are interested in estimating the upper growth rate of ZN, lim supN→∞Z1/NN, that we call the connective constant of the dilute lattice. After proving that this connective constant is a.s.\ non-random, we focus on the two-dimensional case and show that for every percolation parameter p∈(1/2,1), almost surely, ZN grows exponentially slower than its expected value. In other word we prove that $\limsup_{N\to \infty} (Z_N)^{1/N} <\lim_{N\to \infty} \bbE[Z_N]^{1/N}$ where expectation is taken with respect to the percolation process. This result can be considered as a first mathematical attempt to understand the influence of disorder for self-avoiding walk on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on specifics of percolation on $\bbZ^2$, so that our result can be extended to a large family of two dimensional models including general self-avoiding walk in random environment.