Mixing of the Glauber dynamics for the ferromagnetic Potts model |
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| Authors: | Magnus Bordewich Catherine Greenhill Viresh Patel |
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| Affiliation: | 1. School of Engineering and Computing Sciences, Durham University, Durham, UK;2. School of Mathematics and Statistics, The University of New South Wales, Sydney, New South Wales, AustraliaResearch performed while the Catherine Greenhill was on sabbatical at the University of Durham.;3. School of Mathematical Sciences, Queen Mary, University of London, London, UKResearch performed while the Viresh Patel was at the University of Durham. |
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| Abstract: | We present several results on the mixing time of the Glauber dynamics for sampling from the Gibbs distribution in the ferromagnetic Potts model. At a fixed temperature and interaction strength, we study the interplay between the maximum degree (Δ) of the underlying graph and the number of colours or spins (q) in determining whether the dynamics mixes rapidly or not. We find a lower bound L on the number of colours such that Glauber dynamics is rapidly mixing if at least L colours are used. We give a closely‐matching upper bound U on the number of colours such that with probability that tends to 1, the Glauber dynamics mixes slowly on random Δ‐regular graphs when at most U colours are used. We show that our bounds can be improved if we restrict attention to certain types of graphs of maximum degree Δ, e.g. toroidal grids for Δ = 4. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 21–52, 2016 |
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| Keywords: | Glauber dynamics mixing time Potts model ferromagnetic |
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