High-dimensional long-range statistical mechanical models have random walk correlation functions

TL;DR

This paper demonstrates that long-range statistical mechanical models in high dimensions have two-point functions that match random walk correlations, providing bounds near criticality and simplifying lace expansion convergence proofs.

Contribution

It establishes the exact equivalence of two-point functions with random walk models for long-range models above critical dimensions, extending understanding of their critical behavior.

Findings

Two-point functions match random walk correlations up to a constant.

Derived bounds for the two-point function near criticality.

Provided a simplified proof of lace expansion convergence.

Abstract

We consider long-range percolation, Ising model, and self-avoiding walk on $\mathbb{Z}^d$, with couplings decaying like $|x|^{-(d+\alpha)}$ where $0 < \alpha \le 2$, above the upper critical dimensions. In the spread-out setting where the lace expansion applies, we show that the two-point function for each of these models exactly coincides with a random walk two-point function, up to a constant prefactor. Using this, for $0<\alpha < 2$, we prove upper and lower bounds of the form $|x|^{-(d-\alpha)} \min\{ 1, (p_c - p)^{-2} |x|^{-2\alpha} \}$ for the two-point function near the critical point $p_c$. For $\alpha=2$, we obtain a similar upper bound with logarithmic corrections. We also give a simple proof of the convergence of the lace expansion, assuming diagrammatic estimates.