Wednesday, March 28, 2012

1203.6051 (Hubert Lacoin)

On the two dimensional supercritical percolation cluster, the number of
self-avoiding paths is much smaller than expected

Hubert Lacoin
In this paper, we study abundance of self-avoiding paths of a given length on a supercritical percolation percolation cluster for percolation on $\mathbb Z^d$. More precisely, we count $Z_N$ the number of self-avoiding paths of length $N$ on the supercritical cluster, starting from the origin (that we condition to be in the cluster), and are interested in estimating the upper growth rate of $Z_N$ ($\limsup_{N\to \infty} Z_N^{1/N}$, we call it connective constant of the dilute lattice). After proving that the connective constant of the supercritical percolation cluster 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, that is $\limsup_{N\to \infty} Z_N^{1/N}<\lim_{N\to \infty} (\mathbb E[Z_N])^{1/N}$ where expectation is taken with respect to the percolation process. Our method combining change of measure and coarse graining arguments does not rely on specificities of percolation on $\mathbb Z^2$, so that our result can be extended to a large family of two dimensional models including self-avoiding walk in random environment.
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