Talk by Oren Louidor (UCLA)

Title: Mixing time analysis of the Glauber Dynamics for Curie-Weiss Potts Model

Abstract: We consider the discrete-time Glauber dynamics for the q-states Potts model on the complete graph with n vertices and analyze its mixing time, namely the time it takes until the state distribution is close enough to the Potts distribution, starting from the worst possible initial state. In the disorder phase \beta < \beta_c(q), we discover a critical inverse temperature 0 < \beta_d(q) < \beta_c(q) above which mixing time is exponential and below which mixing time is asymptotically C(\beta, q) n \log(n) where C(\beta, q) is known explicitly. In the latter case we prove that the chain exhibits a cutoff with a window of order n, namely after an initial C(\beta, q) n log(n) time the state distribution becomes arbitrarily close to the stationary one in O(n) time. We also analyze the mixing time when the temperature is critical for mixing and more generally when \beta converges to \beta_d(q) with n. In this case the mixing time depends on how fast this convergence is and cutoff appears as long as the rate of convergence is slower then some critical threshold which we identify. In particular in the mixing-critical case we get \Theta(n^{4/3}) mixing time and no cutoff. We also discuss the essential mixing time of the chain and show that it is C(\beta, q) n log(n) for all \beta < \beta_c(q). More precisely, the set of initial states starting from which mixing time is slower than C(\beta, q) n log(n) has exponentially small probability under the Potts distribution.