Potts models and Tutte polynomials

Potts Models and Tutte Polynomials The Ising and Potts models are classic tools in statistical physics, traditionally used to describe magnetic phase transitions. However, their applicability extends far beyond the theory of magnetism. The key to understanding the universality of these models lies in a combinatorial approach to computing their partition functions: up to a certain multiplicative factor, these partition functions are equivalent to Tutte polynomials — fundamental invariants in graph theory. In this overview talk, we will explore how this deep connection is utilized in modern combinatorics and how combinatorial theorems help obtain nontrivial results in physics. In conclusion, we will discuss the possibility of generalizing these results to Ising and Potts models in an external magnetic field. The plan of the talk: 1. Biggs formalism for Potts models2. Combinatorial properties of the partition function of Potts models and their connection with Tutte polynomials3. High-temperature formula and Matiyasevich theorem as the Biggs expansion4. Fortuin–Kasteleyn expansion5. Bracket models, the Jones polynomial as a certain Potts model6. Flow (vertex) models as the image of the discrete Fourier transformation of Potts models7. Kramers–Wannier duality as Tutte duality8. Potts models in an external magnetic field as a Noble–Welsh type polynomial