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

Yucheng Liu

Paper

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

Abstract

We consider long-range percolation, Ising model, and self-avoiding walk on

, with couplings decaying like

where

, 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

, we prove upper and lower bounds of the form

for the two-point function near the critical point

. For

, 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.