On the absence of phase transition in the monomer-dimer model

Abstract

Suppose we cover the set of vertices of a graph $G$ by non-overlapping monomers (singleton sets) and dimers (pairs of vertices corresponding to an edge). Each way to do this is called a monomer-dimer configuration. If $G$ is finite and $lambda > 0$, we define the monomer-dimer distribution for $G$ (with parameter $lambda$) as the probability distribution which assigns to each monomer-dimer configuration a probability proportional to lambda^{|mbox{dimers|, where |mbox{dimers| is the number of dimers in that configuration. If the graph is infinite, monomer-dimer distributions can be constructed in the standard way, by taking weak limits. We are particularly interested in the monomer-dimer model on (subgraphs of) the $d$-dimensional cubic lattice. Heilmann and Lieb (1972) prove absence of phase transition, in terms of smoothness properties of certain thermodynamic functions. They do this by studying the location in the complex plane of the zeros of the partition function. We present a different approach and show, by probabilistic arguments, that boundary effects become negligible as the distance to the boundary goes to $infty$. This gives absence of phase transition in a related, but generally not equivalent sense as above. However, the decay of boundary effects appears to occur in such a strong way that, by results on general Gibbs systems of Dobrushin and Shlosman (1987) and Dobrushin and Warstat (1990), smoothness properties of thermodynamic functions follow. More precisely we show that, in their terminology, the model is {em completely analytic