Order/disorder phase transitions: the example of the Potts model

Abstract

The critical behavior at an order/disorder phase transition has been a central object of interest in statistical physics. In the past century, techniques borrowed from many different fields of mathematics (Algebra, Combinatorics, Probability, Complex Analysis, Spectral Theory, etc) have contributed to a more and more elaborated description of the possible critical behaviors for a large variety of models (interacting particle systems, lattice spin models, spin glasses, percolation models). Through the classical examples of the Ising and Potts models, we survey a few recent advances regarding the rigorous understanding of such phase transitions for the specific case of lattice spin models. This review was written at the occasion of the Harvard/MIT conference Current Developments in Mathematics 2015.