A random walk approach to high-dimensional critical phenomena

TL;DR

This paper introduces a probabilistic, unified approach using random walk techniques to prove mean-field near-critical behavior in high-dimensional lattice models such as percolation and spin systems.

Contribution

It provides a new, simple, and unified proof method for mean-field behavior in high-dimensional models, applicable under broad assumptions.

Findings

Proves decay of two-point functions with specific exponential and polynomial factors.

Applies to multiple models including self-avoiding walk, percolation, and spin systems.

Valid for models above their upper critical dimensions.

Abstract

We present a "black box" proof of mean-field near-critical behaviour for a family of functions on $\mathbb Z^d$ (${d>2}$) satisfying a short list of assumptions. The functions represent two-point functions of a lattice statistical mechanical model in the subcritical or critical regimes, and are proved to have decay of the form $|x|^{-d+2+\varepsilon}\exp[-c|x|/\xi]$, for any $\varepsilon>0$. The black box applies to several models for which commonplace methods can be used to verify the assumptions. Applications include models of self-avoiding walk, percolation, spins (Ising, XY, $|\varphi|^4$), and lattice trees, all above their upper critical dimensions. The proof is based on random walk techniques, and provides a new, unified, probabilistic, and relatively simple proof of mean-field near-critical behaviour.