Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster

Lacoin, Hubert
August 2014
Probability Theory & Related Fields;Aug2014, Vol. 159 Issue 3/4, p777
Academic Journal
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.


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