Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2000-01

January 2000

Random Colorings of a Cayley Tree

Graham R. Brightwell and Peter Winkler


Probability measures on the space of proper colorings of a Cayley tree (that is, an infinite regular connected graph with no cycles) are of interest not only in combinatorics but also in statistical physics, as states of the antiferromagnetic Potts model at zero temperature, on the ``Bethe lattice''.

We concentrate on a particularly nice class of such measures which remain invariant under parity-preserving automorphisms of the tree. Making use of a correspondence with branching random walks on certain bipartite graphs, we determine when more than one such measure exists. The case of ``uniform'' measures is particularly interesting, and as it turns out, plays a special role.

Some of the results herein are deducible from previous work of the authors and by members of the statistical physics community, but many are new. We hope that this work will serve as a helpful glimpse into the rapidly expanding intersection of combinatorics and statistical physics.

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