Bounding the Partition Function of Spin-Systems

With a graph G = (V, E) we associate a collection of non-negative real weights vV {i,v : 1 i m} uvE{ij,uv : 1 i j m}. We consider the probability distribution on {f : V {1, . . . , m}} in which each f occurs with probability proportional to vV f(v),v uvE f(u)f(v),uv. Many well-known statistical physics models, including the Ising model with an external field and the hard-core model with non-uniform activities, can be framed as such a distribution. We obtain an upper bound, independent of G, for the partition function (the normalizing constant which turns the assignment of weights on {f : V {1, . . . , m}} into a probability distribution) in the case when G is a d-regular N-vertex bipartite graph. This generalizes a bound obtained by Galvin and Tetali who considered the simpler weight collection {i : 1 i m} {ij : 1 i j m} with each ij either 0 or 1 and with each f chosen with probability proportional to vV f(v) uvE f(u)f(v). Our main tools are a generalization to list hom...

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: David J. Galvin
• Keywords: COMBINATORICS 2006, Non-negative Real Weights, Probability Distribution, Well-known Statistical Physics