May 2023 Universality of spin correlations in the Ising model on isoradial graphs
Dmitry Chelkak, Konstantin Izyurov, Rémy Mahfouf
Author Affiliations +
Ann. Probab. 51(3): 840-898 (May 2023). DOI: 10.1214/22-AOP1595


We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, that is, as the mesh size tends to zero at the same rate as the Baxter elliptic parameter tends to 1, the two-point spin correlations in the full plane converge to a universal rotationally invariant limit.

These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs, the convergence results for multipoint spin correlations in bounded planar domains which were previously known only on the square grid. We also give a simple proof of the fact that the infinite-volume magnetization in a subcritical Z-invariant Ising model is independent of the site and of the lattice.

As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points which also provides a solid ground for further generalizations.

Funding Statement

D.C. is a holder of the ENS–MHI chair funded by the MHI, whose support is gratefully acknowledged. The work of K.I. was supported by Academy of Finland via academy project “Critical phenomena in dimension two: Analytic and probabilistic methods.” The work of D.C. and R.M. was also partially supported by the ANR-18-CE40-0033 project DIMERS.


We are grateful to S. C. Park for many fruitful discussions of the massive Ising model. We also would like to thank Cédric Boutillier, Béatrice de Tilière, Jhih-Huang Li, Ioan Manolescu, Paul Melotti and Yijun Wan for helpful comments and remarks. Last but not least, we would like to thank the anonymous referee for carefully reading the first version of this paper and providing a useful feedback.


Download Citation

Dmitry Chelkak. Konstantin Izyurov. Rémy Mahfouf. "Universality of spin correlations in the Ising model on isoradial graphs." Ann. Probab. 51 (3) 840 - 898, May 2023.


Received: 1 April 2021; Revised: 1 April 2022; Published: May 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583057
zbMATH: 1517.82012
Digital Object Identifier: 10.1214/22-AOP1595

Primary: 81T40 , 82B20
Secondary: 30G25

Keywords: Planar Ising model , spin correlations , Universality , Z-invariant weights

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 3 • May 2023
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