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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 ($\Delta$) 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 $\Delta$-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 $\Delta$, e.g. toroidal grids for $\Delta=4$.'
author:
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    Magnus Bordewich[^1]\
    School of Engineering and Computing Sciences\
    Durham University\
    Durham, DH1 3LE, UK\
    m.j.r.bordewich@durham.ac.uk\
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    Catherine Greenhill[^2]\
    School of Mathematics and Statistics\
    The University of New South Wales\
    Sydney NSW 2052, Australia\
    csg@unsw.edu.au\
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    Viresh Patel[^3]\
    School of Mathematical Sciences\
    Queen Mary, University of London\
    