Tightness of the Ising–Kac Model on the Two-Dimensional Torus



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 Mourrat and Weber (Commun Pure Appl Math 70:717–812, 2017) and a method by Tsatsoulis and Weber employed in (arXiv:1609.08447 2016), 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 Cassandro et al. (J Stat Phys 78(3):1131–1138, 1995) 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.


Kac potential Ising model Stochastic quantization Glauber dynamic 



We are grateful to H. Weber and P. Tsatsoulis for many discussions on the topic of this article. MH gratefully acknowledges financial support from the Leverhulme trust (Leadership Award) as well as the ERC via the consolidator Grant 615897: CRITICAL.


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Authors and Affiliations

  1. 1.Imperial College LondonLondonUK
  2. 2.University of WarwickCoventryUK

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