A random walk approach to high-dimensional critical phenomena
Abstract
We present a "black box" proof of mean-field near-critical behaviour for a family of functions on () 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 , for any . 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, ), 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.
Cite
@article{arxiv.2605.21438,
title = {A random walk approach to high-dimensional critical phenomena},
author = {Hugo Duminil-Copin and Aman Markar and Romain Panis and Gordon Slade},
journal= {arXiv preprint arXiv:2605.21438},
year = {2026}
}
Comments
85 pages, 7 figures. Corrected a typo in the abstract