Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster
In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on \(\mathbb{Z }^d\). More precisely, we count \(Z_N\), the number of self-avoiding paths of length \(N\) on the infinite cluster starting from the origin (which we condition to be in the cluster). We are interested in estimating the upper growth rate of \(Z_N\), \(\limsup _{N\rightarrow \infty } Z_N^{1/N}\), which 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 \(p\in (1/2,1)\), almost surely, \(Z_N\) grows exponentially slower than its expected value. In other words, we prove that \(\limsup _{N\rightarrow \infty } (Z_N)^{1/N}{<}\lim _{N\rightarrow \infty } \mathbb{E }[Z_N]^{1/N}\), where the 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 walks on a (quenched) dilute lattice. Our method, which combines change of measure and coarse graining arguments, does not rely on the specifics of percolation on \(\mathbb{Z }^2\), so our result can be extended to a large family of two-dimensional models including general self-avoiding walks in a random environment.