Department Colloquium
Department of Mathematics
Princeton University
October 4th, 2000
Hard Constraints and the Bethe Lattice
In recent years an explosion of work in the intersection of
combinatorics and statistical mechanics has contributed lively new
ideas to both areas. Of particular interest to combinatorialists are
physical systems with hard constraints, such as the hard-core gas model
(a.k.a. random independent sets in a graph).
In work with Graham Brightwell of the London School of Economics,
we model hard-constraint systems by the space Hom(G,H) of homomorphisms
from an infinite graph G to a fixed finite constraint graph H. These
spaces become tractable when G is a regular tree (often called a Cayley
tree or Bethe lattice), because the simple, invariant Gibbs measures on
Hom(G,H) then correspond to node-weighted branching random walks on H.
With this approach we can characterize the constraint graphs H which,
by admitting more than one such measure, exhibit phase transitions.
Applications to a physics problem (multiple critical points for
symmetry-breaking) and a combinatorics problem (random coloring) will
be mentioned.