Mathematics Colloquia and Seminars

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Consider a random coloring of a bounded domain in Z^d with the probability of each color configuration proportional to exp(-beta*N(F)), where beta>0 is a parameter (representing the inverse temperature) and N(F) is the number of nearest neighboring pairs colored by the same color. This is the antiferromagnetic 3-state Potts model from statistical physics, used to describe magnetic interactions in a spin system. The Kotecky conjecture is that in such a model, under even constant boundary conditions, for d \ge 3 and high enough beta, a sampled coloring will typically exhibit long-range order. In particular a single color occupies most of either the even or odd vertices of the domain. We give the first rigorous proof of this fact for large d. Our result extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the zero temperature (beta=infinity case). In the talk we shall survey the history and motivation for the model and the conjecture, and describe the evolution of the different probabilistic and combinatorial methods used to tackle such models. No background in statistical physics will be assumed and all terms will be explained thoroughly. Joint work with Yinon Spinka.