Consider a random coloring of a bounded domain in the bipartite graph Z^d with the probability of each color configuration proportional to exp(−β ∗ N(F)), where β > 0 (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, even under constant boundary conditions, for d ≥ 3 and high enough β, a sampled coloring typically exhibits long-range order. In particular a single color occupies most of either the even or odd vertices of the domain. This is in contrast with the situation for small β, when each bipartition class is equally occupied by the three colors.
We give the first rigorous proof of the conjecture for large d. Our result extends previous works of Peled and of Galvin, Kahn, Randall and Sorkin, who treated the zero temperature (β = ∞) case. In this talk we shall give a glimpse into the methods of proof which are combinatorial in nature, relying on structural properties of colorings and odd-boundary subsets of Z^d. No background in statistical physics will be assumed and all terms will be explained thoroughly.
This is joint work with Yinon Spinka.
The Probability Seminars are held in Sequoia Hall, Room 200, at 4:30pm on Mondays. Refreshments are served at 4pm in the Lounge on the first floor.