Mixing Time Power Laws at Criticality

We study the mixing time of some Markov chains converging to critical physical models. These models are indexed by a parameter beta and there exists some critical value beta_c where the model undergoes a phase transition. According to physics lore, the mixing time of such Markov chains is often of logarithmic order outside the critical regime, when beta ne beta_c, and satisfies-some power law at criticality, when beta = beta_c. We prove this in the two following settings: 1. Lazy random walk on the critical percolation cluster of "mean-field" graphs, which include the complete graph and random d-regular graphs. The critical mixing time here is of order Theta(n). This answers a question of Benjamini, Kozma and Wormald. 2. Swendsen-Wang dynamics on the complete, graph. The critical mixing time, here is of order Theta(n^1/4). This improves results of Cooper, Dyer, Frieze and Rue. In both settings, the main tool is understanding the Markov chain dynamics via properties of critical percolation on the underlying graph.