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

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

2005-04-06

arxiv.org/abs/math-ph/0504021v3

Annals of Probability 2008, Vol. 36, No. 2, 530-593

Physics

Mathematical Physics

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

Scientific paper

10.1214/009117907000000231

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 $|x|\to\infty$.

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