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

Probability Seminar

16th February 2022, 4:00 pm – 5:00 pm

Online,

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.

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