Statistical mechanics on isoradial graphs

TL;DR

This survey explores how isoradial graphs serve as a natural framework for studying critical phenomena in various statistical mechanics models, connecting discrete complex analysis with physical theories.

Contribution

It provides a comprehensive overview of the role of isoradial graphs in statistical mechanics, linking physical approaches with mathematical structures and summarizing key results.

Findings

Explicit results for the critical dimer model on bipartite isoradial graphs

Analysis of the 2D critical Ising model on isoradial graphs

Insights into random walk, spanning trees, and Potts model on these graphs

Abstract

Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs naturally arise in two approaches used by physicists: transfer matrices and conformal field theory. This leads us to the fact that isoradial graphs provide a natural setting for discrete complex analysis, to which we dedicate one section. Then, we give an overview of explicit results obtained for different models of statistical mechanics defined on such graphs: the critical dimer model when the underlying graph is bipartite, the 2-dimensional critical Ising model, random walk and spanning trees and the q-state Potts model.