The Potts and random-cluster models

Abstract

The relationship between the Tutte polynomial and the random-cluster, Ising, and Potts models of statistical physics is summarized. Certain fundamental properties of these models are described, particularly those that may be expressed neatly in terms of Tutte polynomials. Ising, Potts, and random-cluster models; physical origins; couplings; partition functions. Basic properties of random-cluster measures, stochastic ordering, comparison inequalities, positive association. Limit as https://www.w3.org/1998/Math/MathML"> q ↓ 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429161612/ab6e6aa5-46ff-432c-83f4-7ad3e3757dc3/content/math20_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , uniform spanning tree, uniform spanning forest, uniform connected subgraph, negative association. Flow polynomial, Potts two-point correlation, Simon inequality. Zero-temperature limit and the chromatic polynomial. Asymptotics of the Tutte polynomial on the complete graph.