Asymptotics of even-even correlations in the Ising model

TL;DR

This paper investigates the asymptotic behavior of correlations in finite-range ferromagnetic Ising models below the critical temperature, focusing on the decay of covariances between even-sized sets as their separation grows.

Contribution

It provides a detailed analysis of the prefactor to the exponential decay of correlations in the Ising model, extending understanding of correlation decay in the subcritical regime.

Findings

Derived asymptotic formulas for covariance decay

Identified the behavior of the prefactor in correlation decay

Extended results to arbitrary finite sets of even cardinality

Abstract

We consider finite-range ferromagnetic Ising models on $\mathbb{Z}^d$ in the regime $\beta<\beta_c$. We analyze the behavior of the prefactor to the exponential decay of $\mathrm{Cov}(\sigma_A,\sigma_B)$, for arbitrary finite sets $A$ and $B$ of even cardinality, as the distance between $A$ and $B$ diverges.