## Abstract

Consider the nearest-neighbor Ising model on Λ _{n}: = [- n, n] ^{d}∩ Z^{d} at inverse temperature β≥ 0 with free boundary conditions, and let Yn(σ):=∑u∈Λnσu be its total magnetization. Let X_{n} be the total magnetization perturbed by a critical Curie–Weiss interaction, i.e., dFXndFYn(x):=exp[x2/(2⟨Yn2⟩Λn,β)]〈exp[Yn2/(2⟨Yn2⟩Λn,β)]〉Λn,β,where FXn and FYn are the distribution functions for X_{n} and Y_{n} respectively. We prove that for any d≥ 4 and β∈ [0 , β_{c}(d)] where β_{c}(d) is the critical inverse temperature, any subsequential limit (in distribution) of {Xn/E(Xn2):n∈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 β small, fX(x)=Kexp(-C4x4),where C=Γ(3/4)/Γ(1/4) and K=Γ(3/4)/(4Γ(5/4)3/2). Possible connections between f_{X} and the high-dimensional critical Ising model with periodic boundary conditions are discussed.

Original language | English (US) |
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Article number | 5 |

Journal | Journal of Statistical Physics |

Volume | 188 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2022 |

## Keywords

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

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics