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

Martin Hairer; Massimo Iberti

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arXiv · 1712.08678

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

Martin Hairer·Massimo Iberti

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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.

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Martin Hairer, Massimo Iberti. 2017-12-22. Tightness of the Ising-Kac model on the two-dimensional torus. https://arxiv.org/abs/1712.08678

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