Ising Model with Curie-Weiss Perturbation

作者：Camia, Federico; Jiang, Jianping^*; Newman, Charles M.

来源：Journal of Statistical Physics, 2022, 188(1): 5.

DOI：10.1007/S10955-022-02935-1

• Ising model
• Curie-Weiss interaction
• Analytic density
• Pure imaginary zeros
• High dimensions
• Periodic boundary conditions

Consider the nearest-neighbor Ising model on Lambda(n) := [-n, n](d) boolean AND Z(d) at inverse temperature beta >= 0 with free boundary conditions, and let Y-n (sigma) := Sigma(u is an element of Lambda n) sigma(u) be its total magnetization. Let X-n be the total magnetization perturbed by a critical Curie-Weiss interaction, i.e., @@@ d FXn/d FYn (x) := exp[x(2)/(2 < Y-n(2)>(Lambda n, beta))]/< exp[Y-n(2)/ (2 < Y-n(2)>(Lambda n, beta))]>(Lambda n, beta), @@@ where F-Xn and F-Yn are the distribution functions for X-n and Y-n respectively. We prove that for any d >= 4 and beta is an element of [0, beta(c)(d)] where beta(c)(d) is the critical inverse temperature, any subsequential limit (in distribution) of {X-n/root E (X-n(2)) : n is an element of 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 >= 1 and then for beta small, @@@ f(X)(x) = K exp(-C(4)x(4)), @@@ where C = root Gamma(3/4)/Gamma(1/4) and K = root 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.

• 单位
  清华大学