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

TL;DR

This paper proves that the two-point functions for various lattice models decay as a constant times |x|^{2-d} at criticality in high dimensions, using the lace expansion technique.

Contribution

It establishes the asymptotic decay of two-point functions for self-avoiding walk, percolation, lattice trees, and animals in high dimensions, extending previous spread-out model results.

Findings

Two-point functions decay as |x|^{2-d} at criticality in high dimensions.

Derived sharp conditions for the asymptotic behavior of random walk two-point functions.

Extended decay results to models with nearest-neighbor interactions in specific high dimensions.

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 $\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$, for $d\geq 5$ for self-avoiding walk, for $d\geq19$ 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 $\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$.