English

A random walk approach to high-dimensional critical phenomena

Probability 2026-05-22 v2 Mathematical Physics math.MP

Abstract

We present a "black box" proof of mean-field near-critical behaviour for a family of functions on Zd\mathbb Z^d (d>2{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 xd+2+εexp[cx/ξ]|x|^{-d+2+\varepsilon}\exp[-c|x|/\xi], for any ε>0\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, φ4|\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.

Keywords

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

R2 v1 2026-07-22T07:24:27.911Z