Ising Model with Curie–Weiss Perturbation

Abstract

Consider the nearest-neighbor Ising model on $$\Lambda _n:=[-n,n]^d\cap {\mathbb {Z}}^d$$ at inverse temperature $$\beta \ge 0$$ 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., $$\begin{aligned} \frac{d F_{X_n}}{d F_{Y_n}}(x):=\frac{\exp [x^2/\left( 2\langle Y_n^2 \rangle _{\Lambda _n,\beta }\right) ]}{\left\langle \exp [Y_n^2/\left( 2\langle Y_n^2\rangle _{\Lambda _n,\beta }\right) ]\right\rangle _{\Lambda _n,\beta }}, \end{aligned}$$ where $$F_{X_n}$$ and $$F_{Y_n}$$ are the distribution functions for $$X_n$$ and $$Y_n$$ respectively. We prove that for any $$d\ge 4$$ 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 $$d\ge 1$$ and then for $$\beta $$ small, $$\begin{aligned} f_X(x)=K\exp (-C^4x^4), \end{aligned}$$ 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.