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Can extra updates delay mixing? (English) Zbl 1277.82036
Summary: We consider Glauber dynamics (starting from an extremal configuration) in a monotone spin system, and show that interjecting extra updates cannot increase the expected Hamming distance or the total variation distance to the stationary distribution. We deduce that, for monotone Markov random fields, when block dynamics contracts a Hamming metric, single-site dynamics mixes in \(O(n \log n)\) steps on an \(n\)-vertex graph. In particular, our result completes work of Kenyon, Mossel and Peres concerning Glauber dynamics for the Ising model on trees. Our approach also shows that on bipartite graphs, alternating updates systematically between odd and even vertices cannot improve the mixing time by more than a factor of \(\log n\) compared to updates at uniform random locations on an \(n\)-vertex graph. Our result is especially effective in comparing block and single-site dynamics; it has already been used in works of Martinelli, Toninelli, Sinclair, Mossel, Sly, Ding, Lubetzky, and Peres in various combinations.

MSC:
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82D40 Statistical mechanical studies of magnetic materials
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G60 Random fields
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