Department Colloquium
Department of Mathematics
Princeton University
April 5th, 2000
Random Colorings of a Cayley Tree
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. Using branching random walks, we determine when more than one such measure exists. This talk (on joint work with Graham Brightwell, of the London School of Economics) will provide, we hope, a helpful glimpse into the rapidly expanding intersection of combinatorics and statistical physics.