A Cluster Expansion and the Decay of Correlations of the 1D Long-Range Ising Model at Low Temperatures

TL;DR

This paper develops a convergent low-temperature cluster expansion for the 1D long-range Ising model with polynomial decay, revealing that correlations decay algebraically with a rate equal to the decay exponent.

Contribution

It introduces a new low-temperature cluster expansion for the 1D long-range Ising model with polynomial decay, providing precise correlation decay rates.

Findings

Two-point correlations decay algebraically with rate 

The cluster expansion converges at low temperatures for 

Explicit relation between decay exponent and correlation decay rate

Abstract

In this work, a convergent low-temperature cluster expansion of the one-dimensional long-range ferromagnetic Ising model with polynomial decay $\alpha\in (1,2]$ is developed; that is, $J(r)=r^{-\alpha}$. As an application, the $n$-point correlations are studied and the two-point correlation is shown to be algebraic with rate of decay exactly $\alpha$.