Current Developments in Mathematics

Volume 2015

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

Pages: 27 – 71

DOI: http://dx.doi.org/10.4310/CDM.2015.v2015.n1.a2

Author

Hugo Duminil-Copin (Département de Mathématiques, Université de Genève, Switzerland)

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.

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