We study the q-state ferromagnetic Potts model on random regular graphs. It is conjectured that the model exhibits metastability phenomena, i.e., the presence of “phases” (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape. In this talk, I will detail the emergence of the two relevant phases for the q-state Potts model on the d-regular random graph for all integers q,d >= 3. The proofs are based on a conceptual connection between spatial properties and the structure of the Potts distribution on the random regular graph, rather than complicated moment calculations. I will also discuss consequences on relevant Markov chains (both local and non-local) and recent approaches to work around the metastability.

Joint with A. Coja-Oghlan, L.A. Goldberg, J.B. Ravelomanana, D. Stefankovic, P. Smolarova, and E. Vigoda.