Tightness of the Ising-Kac model on the two-dimensional torus

TL;DR

This paper proves tightness and characterizes the limiting measure of the Ising-Kac model on a two-dimensional torus near criticality, using dynamic methods instead of correlation inequalities, covering the entire critical regime.

Contribution

It introduces a dynamic-based approach to analyze the Ising-Kac model on the torus, avoiding previous assumptions and extending results to the full critical regime.

Findings

Proves tightness of Gibbs measures near criticality

Characterizes the limit as the $\

law of the $\

Abstract

We consider the sequence of Gibbs measures of Ising models with Kac interaction defined on a periodic two-dimensional discrete torus near criticality. Using the convergence of the Glauber dynamic proven by H. Weber and J.C. Mourrat and a method by H. Weber and P. Tsatsoulis, we show tightness for the sequence of Gibbs measures of the Ising-Kac model near criticality and characterise the law of the limit as the $\Phi^4_2$ measure on the torus. Our result is very similar to the one obtained by M. Cassandro, R. Marra and E. Presutti on $\mathbb{Z}^2$, but our strategy takes advantage of the dynamic, instead of correlation inequalities. In particular, our result covers the whole critical regime and does not require the large temperature / large mass / small coupling assumption present in earlier results.