Ising model with Curie-Weiss perturbation

TL;DR

This paper investigates the asymptotic distribution of the magnetization in a high-dimensional Ising model with a Curie-Weiss perturbation, revealing explicit forms and properties of the limiting density function.

Contribution

It provides a detailed analysis of the limiting distribution of the perturbed magnetization in high-dimensional Ising models, including explicit density expressions and zero distributions of the moment generating function.

Findings

Limit distributions have analytic densities with purely imaginary zeros.

Explicit density form for small beta: Gaussian-like with a quartic exponential decay.

Connections discussed between the limiting distribution and high-dimensional critical Ising models.

Abstract

Consider the nearest-neighbor Ising model on $\Lambda_n:=[-n,n]^d\cap\mathbb{Z}^d$ at inverse temperature $\beta\geq 0$ with free boundary conditions, and let $Y_n(\sigma):=\sum_{u\in\Lambda_n}\sigma_u$ be its total magnetization. Let $X_n$ be the total magnetization perturbed by a critical Curie-Weiss interaction, i.e., \begin{equation*} \frac{d F_{X_n}}{d F_{Y_n}}(x):=\frac{\exp[x^2/\left(2\langle Y_n^2 \rangle_{\Lambda_n,\beta}\right)]}{\left\langle\exp[Y_n^2/\left(2\langle Y_n^2\rangle_{\Lambda_n,\beta}\right)]\right\rangle_{\Lambda_n,\beta}}, \end{equation*} where $F_{X_n}$ and $F_{Y_n}$ are the distribution functions for $X_n$ and $Y_n$ respectively. We prove that for any $d\geq 4$ and $\beta\in[0,\beta_c(d)]$ where $\beta_c(d)$ is the critical inverse temperature, any subsequential limit (in distribution) of $\{X_n/\sqrt{\mathbb{E}\left(X_n^2\right)}:n\in\mathbb{N}\}$ has an analytic density (say, $f_X$) all of whose zeros are pure imaginary, and $f_X$ has an explicit expression in terms of the asymptotic behavior of zeros for the moment generating function of $Y_n$. We also prove that for any $d\geq 1$ and then for $\beta$ small, \begin{equation*} f_X(x)=K\exp(-C^4x^4), \end{equation*} where $C=\sqrt{\Gamma(3/4)/\Gamma(1/4)}$ and $K=\sqrt{\Gamma(3/4)}/(4\Gamma(5/4)^{3/2})$. Possible connections between $f_X$ and the high-dimensional critical Ising model with periodic boundary conditions are discussed.