Improved Peierls Argument for High Dimensional Ising Models

TL;DR

This paper improves the understanding of the low temperature behavior of high-dimensional Ising models by establishing the convergence of the Peierls expansion at temperatures proportional to d divided by log d, matching the expected order.

Contribution

The paper provides a refined convergence result for the Peierls expansion in high-dimensional Ising models, extending the temperature range where the expansion is valid.

Findings

Convergence of the Peierls expansion for all temperatures below C d / log d.

The result matches the expected order in the dimension d.

Improves previous bounds on the temperature range for convergence.

Abstract

We consider the low temperature expansion for the Ising model on $\Z^d$, $d \ge 2$, with ferromagnetic nearest neighbor interactions in terms of Peierls contours. We prove that the expansion converges for all temperatures smaller than $C d (\log d)^{-1}$, which is the correct order in $d$.