Rapid mixing of Gibbs samplers: Coupling, Spectral Independence, and Entropy factorizations.

We discuss some recent developments in the analysis of convergence to stationarity for the Gibbs sampler of general spin systems on arbitrary graphs. These are based on two recently introduced concepts: Spectral Independence and Block Factorization of Entropy. We show that if a system is spectrally independent then its entropy functional satisfies a general block factorization, which in turn implies a modified log-Sobolev inequality and a tight control of the mixing time for the Glauber dynamics as well as for any other heat bath block dynamics. Moreover, we show that the existence of a contractive coupling for a local Markov chain implies that the system is spectrally independent. As a corollary, we obtain new optimal bounds on the mixing time of a large class of sampling algorithms for the ferromagnetic Ising/Potts models in the so-called tree-uniqueness regime, including non-local Markov chains such as the Swendsen-Wang dynamics. The methods also apply to spin systems with hard constraints such as q-colorings of a graph and the hard-core gas. Based on some recent joint works with Antonio Blanca, Zongchen Chen, Daniel Parisi, Alistair Sinclair, Daniel Stefankovic, and Eric Vigoda.