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 Z d . The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x ∈ 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| → ∞, for d ≥ 5 for self-avoiding walk, for d ≥ 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 Z d is asymptotic to const.|x| 2−d as |x| → ∞.

Enlargement of subgraphs of infinite graphs by Bernoulli percolation

“…verifying several conditions in Hara's paper [16], which shows this decay rate for d ≥ 19. Therefore,…”

Mean-Field Behavior for Long- and Finite Range Ising Model, Percolation and Self-Avoiding Walk

“…A general argument that holds for all our three models is the following: the quantity G z (x) can be realized as an increasing limit (finite volume approximation) of a function which is continuous and non-decreasing in z, hence G z (x) is left-continuous (cf. [25,Appendix A]). It follows that (1.38)-(1.40) even hold at criticality, i.e.…”

“…Partial results towards (1.44) have been obtained. Indeed, Hara, van der Hofstad and Slade [29] proved (1.44) in the finite-range spread-out setting for self-avoiding walk and percolation, Hara [25] proved it in the nearest-neighbor setting, and Sakai [38] proved it for the Ising model in finite-range spread-out and nearest-neighbor settings. We discuss the critical two-point function G zc (x) at the end of Sect.…”

Arm exponents in high dimensional percolation

“…The second estimate that we use, derived by Hara [17] (for the nearest-neighbor model) and by Hara, van der Hofstad and Slade [18] (for the spread-out model) states that in high dimensions,…”