Circle Patterns and Critical Ising Models

Marcin Lis

doi:10.1007/s00220-019-03541-1

Abstract

A circle pattern is an embedding of a planar graph in which each face is inscribed in a circle. We define and prove magnetic criticality of a large family of Ising models on planar graphs whose dual is a circle pattern. Our construction includes as a special case the critical isoradial Ising models of Baxter.

Highlights

The exact value of critical parameters is known only for a limited number of twodimensional models of statistical mechanics
One of the most prominent examples is the Ising model whose critical temperature on the square lattice was famously computed by Kramers and Wannier [24] under a uniqueness hypothesis on the critical point
The critical point in the case of the square lattice with periodic coupling constants was first computed by Li [25], and the result was later extended to all biperiodic graphs by Cimasoni and Duminil-Copin [14]

Summary

Introduction

The exact value of critical parameters is known only for a limited number of twodimensional models of statistical mechanics. The critical point in the case of the square lattice with periodic coupling constants was first computed by Li [25], and the result was later extended to all biperiodic graphs by Cimasoni and Duminil-Copin [14] Both approaches go through Fourier analysis of periodic matrices that arise from the combinatorial solutions of the Ising model due to Fisher [19], and Kac and Ward [22] respectively. In this article we define coupling constants for the Ising model on the dual graph of a circle pattern, and prove that the resulting model is critical. The foundation for the proof of conformal invariance on isoradial graphs is the strong form of discrete holomorphicity that is satisfied by the fermionic observable associated with the critical Ising model.

Main Results

Proofs of Main Results

Je sinh 2 Je

Discrete Holomorphicity of Fermionic Observables

Applications to Other Graphs