Universality of Ising spin correlations on critical doubly-periodic graphs

TL;DR

This paper proves that Ising spin correlations on critical doubly periodic graphs exhibit conformal invariance in the scaling limit, confirming universality across different lattice structures beyond the square lattice.

Contribution

It extends the universality of Ising model correlations to doubly periodic graphs by combining discrete analytic techniques with random cluster methods, overcoming previous limitations.

Findings

Conformal invariance of Ising correlations on doubly periodic graphs.

Scaling limits match those of the critical square lattice.

Completes the universality picture for periodic lattices.

Abstract

We establish conformal invariance of Ising spin correlations on critical doubly periodic graphs, showing that their scaling limit coincides with that of the critical square lattice, as originally proved by Chelkak, Hongler and Izyurov. To overcome the absence of integrability and quantitative full plane constructions in the periodic setting, we combine discrete analytic tools with random cluster methods. This result completes the universality picture for periodic lattices, whose criticality condition was identified by Cimasoni and Duminil-Copin and whose conformal structure and interface convergence were obtained by Chelkak.