This week

A common challenge in using Markov chain for sampling from high-dimensional distributions is multimodality, where the chain may get trapped far from stationarity. However, this issue often applies only to worst-case initializations and can be mitigated by using high-entropy initializations, such as product or weakly correlated distributions. From such starting points, the dynamics can escape saddle points and spread mass correctly across dominant modes.

In this talk, I will discuss our results on convergence from high-entropy initializations for the random-cluster models on the complete graph. We focus on the Chayes–Machta or the Swendsen–Wang dynamics for the random-cluster model showing that these chains mix rapidly from specific product measures, even though they mix exponentially slowly from worst-case initializations. The analogous results hold for the Glauber dynamics on the Potts model. Our proofs involve approximating high-dimensional dynamics with 1-dimensional random processes and analyzing their escape from saddle points.