Berry--Esseen Bounds for High-Temperature Ising Models

Abstract

We consider the nearest-neighbour Ising model on $\mathbb{Z}^d$, and establish sharp results for the normal approximation of the magnetisation throughout the high-temperature regime. In particular, we establish optimal Berry--Esseen bounds between the magnetisation in an arbitrary finite region and a Gaussian with the corresponding finite-volume variance. These results hold for finite-volume Gibbs measures with arbitrary boundary conditions, as well as for the infinite-volume Gibbs measure. We also determine sharp rates of convergence of the magnetisation in a finite box to a Gaussian with variance given by the infinite-volume susceptibility. For this fixed Gaussian target, we show that, in dimensions $d\ge 3$, periodic boundary conditions yield strictly faster convergence than free boundary conditions. Our approach uses Stein's method to quantify the normal approximation error by analysing the semigroup of the continuous-time Ising heat-bath dynamics.