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 $\mathbb Z^d$ (${d>2}$) 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.