Normal approximation for total magnetization in subcritical long-range Ising models

Abstract

We establish Berry--Esseen bounds and 1-Wassershtein bounds for the total spin of strictly subcritical zero-field Ising models with summable interactions. Let $J:{\mathbb{Z}}^d\to [0,\infty )$ be symmetric, satisfy $J(0)=0$, and $\Vert J{\Vert }_1:={\sum }_{x\in {\mathbb{Z}}^d}J(x)<\infty$. For a nonempty finite set $\Lambda \subset {\mathbb{Z}}^d$, consider zero-field Gibbs measure with free boundary conditions, in which a spin configuration $\sigma \in \{-1,+1{\}}^{\Lambda }$ has probability proportional to \[ \exp\!\left\{ β\sum_{x,y\inΛ} J(x-y)σ_xσ_y \right\}. \] Let $S_{\Lambda }:={\sum }_{x\in \Lambda }{\sigma }_x$ be the total spin and ${\beta }_{\mathrm{c}}$ denote the critical inverse temperature. For every fixed $0\le \beta <{\beta }_{\mathrm{c}}$, we prove that \[ d_{\mathrm K}\!\left( \frac{S_Λ}{\sqrt{\operatorname{Var}(S_Λ)}},G \right) \leq C_{β,J}|Λ|^{-1/2}, \] where $d_{\mathrm{K}}$ is the Kolmogorov distance, $G\sim \mathcal{N}(0,1)$, and $C_{\beta ,J}$ depends only on $\beta$ and $J$. For the fully periodized model on $(\mathbb{Z}/L\mathbb{Z}{)}^d$, the corresponding rate is $L^{-d/2}$, uniformly in $L$. For the classical nearest-neighbor model, the Kolmogorov bound also holds for $-{\beta }_{\mathrm{c}}<\beta <{\beta }_{\mathrm{c}}$, covering both ferromagnetic and antiferromagnetic interactions. We also obtain $1$-Wasserstein bounds with the same rates in the respective subcritical regimes. Both the Kolmogorov and $1$-Wasserstein bounds hold with the same rate under the corresponding infinite-volume free-boundary Gibbs measures.

arXiv comment48 pages