2012/06/14 by Galvin, David
#05C15 #82B20 #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1206.3200
With a graph G=(V,E) we associate a collection of non-negative real weights ∪v∈ Vλi,v:1≤ i ≤ m ∪ ∪uv ∈ E λij,uv:1≤ i ≤ j ≤ m. We consider the probability distribution on f:V→1,...,m in which each f occurs with probability proportional to ∏v ∈ Vλf(v),v∏uv ∈ Eλf(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 regular bipartite graph. This generalizes a bound obtained by Galvin and Tetali who considered the simpler weight collection \λi:1 ≤ i ≤ m\ ∪ \λij:1 ≤ i ≤ j ≤ m\ with each λij either 0 or 1 and with each f chosen with probability proportional to ∏v ∈ Vλf(v)∏uv ∈ Eλf(u)f(v). Our main tools are a generalization to list homomorphisms of a result of Galvin and Tetali on graph homomorphisms and a straightforward second-moment computation.