ENERGY CORRELATIONS IN THE CRITICAL ISING MODEL ON A TORUS

• 成果类型: Article
• 署名作者: Izyurov, Konstantin; Kemppainen, Antti; Tuisku, Petri
• 署名单位: University of Helsinki
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1968
• 发表日期: 2024
• 页码: 1699-1729
• 关键词: size-scaling corrections; crystal statistics; conformal-invariance; dimers; range; plane; ORDER

摘要

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(EV -EH) of the energy density on a vertical and a horizontal edge, which is of order of 82, where 8 is the mesh size. The observable EV - EH 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.