A random walk approach to high-dimensional critical phenomena

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 $\mathrm{\mid }x{\mathrm{\mid }}^{-d+2+\epsilon }\exp [-c\mathrm{\mid }x\mathrm{\mid }\mathrm{/}\xi ]$, for any $\epsilon >0$. 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, $\mathrm{\mid }\phi {\mathrm{\mid }}^4$), 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.