Long range order in random three-colorings of Z^d

Series: 
Combinatorics Seminar
Friday, April 15, 2016 - 15:00
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
Stanford University
Organizer: 

Joint work with Yinon Spinka.

Consider a random coloring of a bounded domain in the bipartite graph Z^d with the probability of each color configuration proportional to exp(-beta*N(F)), where beta>0, and N(F) is the number of nearest neighboring pairs colored by the same color. This model of random colorings biased towards being proper, 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 with d >= 3, Fixing the boundary of a large even domain to take the color $0$ and high enough beta, a sampled coloring would 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 beta, 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 beta=infinity case, where the coloring is chosen uniformly for all proper three-colorings. In the talk we shell give a glimpse into the combinatorial methods used to tackle the problem. These rely on structural properties of odd-boundary subsets of Z^d. No background in statistical physics will be assumed and all terms will be thoroughly explained.