Abstract:

Let $G=(V,E)$ be an undirected graph with vertices $V$ and edges $E$. A dimer is a domino occupying an edge $e=(u,v) \in E$ and a monomer is a single vertex $w\in V$. An $l$-match, or a monomer-dimer cover of $G$, is a subset $E´$ of $E$, of cardinality $l$, such that any two distinct edges $e,f\in E´$ do not have a common vertex. Let $\phi(l,G) \ge 0$ be the number of $l$-matchings in $G$ for any $l \in \mathbb{Z}_+$. In many combinatorial problems it is of interest to estimate from above and below the number $\phi(l,G)$. In the monomer-dimer models in statistical mechanics, as the integer lattice $\mathbb{Z}^d$, or the Bethe lattice, i.e. an infinite $k$-regular tree, one needs to estimate the number of $l$-matchings in bipartite regular graphs. Moreover, one needs to estimate the corresponding quantities $h_G(p)$, called the monomer-dimer entropy, for dimer density $p\in [0,1]$, for infinite graphs $G$. In this lecture we will survey the recent developments in this area and pose several conjectures.

The slides of the lecture are available at http://www.math-berlin.de/images/stories/friedland.pdf