Cluster Expansion and Decay of Correlations for Multidimensional Long-Range Ising Models

TL;DR

This paper develops a cluster expansion method for multidimensional long-range Ising models with algebraic decay of interactions, demonstrating how correlations decay with distance at low temperatures.

Contribution

It introduces a new cluster expansion framework for long-range Ising models with power-law interactions, enabling analysis of correlation decay.

Findings

Correlation functions decay algebraically with exponent t at low temperatures

Cluster expansion converges for models with t > d

Provides insights into phase behavior of long-range systems

Abstract

We develop the cluster expansion for the multidimensional multiscaled contours defined by three of us. These contours are suitable for long-range Ising models with interaction $J_{xy}=J(|x-y|)= J/|x-y|^\alpha$, $J>0$, and $\alpha>d$. As an application of the convergence of the cluster expansion at low temperatures, we study the decay of the truncated two-point correlation functions, showing that the decay is algebraic with coefficient $\alpha$.