Georgia Institute of Technology College of Sciences

The Potts model is a distribution on q-colorings of a graph, used to represent spin configurations of a system of particles. Intuitively we expect most configurations to be "solid-like" at low temperatures and "gas-like" at high temperatures. We prove a precise version of this statement for d-regular expander graphs. We also consider the question of whether or not there are efficient algorithms for approximate counting and sampling from the model, and show that such algorithms exist at almost all temperatures. In this talk, I will introduce the different tools we use in our proofs, which come from both statistical physics (polymer models, cluster expansion) and combinatorics (a new container-like result, Karger's randomized min-cut algorithm). This is joint work with Charlie Carlson, Ewan Davies, Nicolas Fraiman, Alexandra Kolla, and Aditya Potukuchi.