April 2024 Energy correlations in the critical Ising model on a torus
Konstantin Izyurov, Antti Kemppainen, Petri Tuisku
Author Affiliations +
Ann. Appl. Probab. 34(2): 1699-1729 (April 2024). DOI: 10.1214/23-AAP1968


We compute rigorously the scaling limit of multipoint energy correlations in the critical Ising model on a torus. For the one-point function, averaged between horizontal and vertical edges of the square lattice, this result has been known since the 1969 work of Ferdinand and Fischer. We propose an alternative proof, in a slightly greater generality, via a new exact formula in terms of determinants of discrete Laplacians. We also compute the main term of the asymptotics of the difference E(ϵVϵH) of the energy density on a vertical and a horizontal edge, which is of order of δ2, where δ is the mesh size. The observable ϵVϵH has been identified by Kadanoff and Ceva as (a component of) the stress-energy tensor.

We then apply the discrete complex analysis methods of Smirnov and Hongler to compute the multipoint correlations. The fermionic observables are only periodic with doubled periods; by antisymmetrization, this leads to contributions from four “sectors.” The main new challenge arises in the doubly periodic sector, due to the existence of nonzero constant (discrete) analytic functions. We show that some additional input, namely the scaling limit of the one-point function and of relative contribution of sectors to the partition function, is sufficient to overcome this difficulty and successfully compute all correlations.

Funding Statement

Work supported by the Academy of Finland via Centre of Excellence in Analysis and Dynamics Research and the academy project “Critical phenomena in dimension two: analytic and probabilistic methods.”


We are grateful to Antti Kupiainen and Dmitry Chelkak for useful discussions, and to the anonymous referee for careful reading of the manuscript and many useful suggestions. We thank David Loeffler for pointing out a quick way to derive the Kronecker limit formula with antiperiodic boundary conditions, used in the proof of Corollary 2.


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Konstantin Izyurov. Antti Kemppainen. Petri Tuisku. "Energy correlations in the critical Ising model on a torus." Ann. Appl. Probab. 34 (2) 1699 - 1729, April 2024. https://doi.org/10.1214/23-AAP1968


Received: 1 April 2021; Revised: 1 March 2023; Published: April 2024
First available in Project Euclid: 3 April 2024

MathSciNet: MR4727677
Digital Object Identifier: 10.1214/23-AAP1968

Primary: 82B20 , 82B27

Keywords: Critical phenomena , discrete holomorphicity , Ising model

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.34 • No. 2 • April 2024
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