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

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

A correlation inequality for connection events in percolation

It is well-known in percolation theory (and intuitively plausible) that two events of the form ``there is an open path from $s$ to $a$' are positively correlated. We prove the (not intuitively obvious) fact that this is still true if we condition on an event of the form ``there is no open path from $s$ to $t$'

Asymptotic density in a coalescing random walk model

We consider a system of particles, each of which performs a continuous time random walk on {bf Z^d. The particles interact only at times when a particle jumps to a site at which there are a number of other particles present. If there are $j$ particles present, then the particle which just jumped is removed from the system with probability $p_j$. We show that if $p_j$ is increasing in $j$ and if the dimension $d$ is at least 6 and if we start with one particle at each site of {bf Z^d, then p(t) := P{there is at least one particle at the origin at time $t sim C(d)/t$. The constant $C(d)$ is explicitly identified. We think the result holds for every dimension $d ge 3$ and we briefly discuss which steps in our proof need to be sharpened to weaken our assumption $d ge 6$. The proof is based on a justification of a certain mean field approximation for $dp(t)/dt$. The method seems applicable to many more models of coalescing and annihilating particles