Statistical mechanics on isoradial graphs

Cédric Boutillier (1) , Béatrice de Tilière (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: statistical mechanics; isoradial graphs; Z-invariant models; Yang-Baxter equations; discrete complex analysis; dimers; critical Ising model; statistical mechanics

Domaines: Probabilités [math.PR]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Cédric Boutillier, Béatrice de Tilière. Statistical mechanics on isoradial graphs. Jean-Dominique Deuschel, Barbara Gentz, Wolfgang König, Max von Renesse, Michael Scheutzow, Uwe Schmock. Probability in Complex Physical Systems, 11, Springer, pp.491-512, 2012, Springer Proceedings in Mathematics, ⟨10.1007/978-3-642-23811-6_20⟩. ⟨hal-00546224⟩