Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals

Abstract

We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ${\mathbb{Z}}^d$. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to $x\in {\mathbb{Z}}^d$, the probability of a connection from the origin to $x$, and the generating functions for lattice trees or lattice animals containing the origin and $x$. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to $const.|x{|}^{2-d}$ as $|x|\to \infty$, for $d\ge 5$ for self-avoiding walk, for $d\ge 19$ for percolation, and for sufficiently large $d$ for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349--408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if $d>4$) condition under which the two-point function of a random walk on ${\mathbb{Z}}^d$ is asymptotic to $const.|x{|}^{2-d}$ as $|x|\to \infty$.

Published in at http://dx.doi.org/10.1214/009117907000000231 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• On the Number of Incipient Spanning Clusters

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