COLLOQUIUM
3:00 p.m., Friday (March 6, 2009)
MATX 1100
Shmuel Friedland
University of Illinois, Chicago
Counting matchings in graphs with applications to the monomerdimer models
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 lmatch, or a monomerdimer cover of G, is a subset E' of E, of cardinality l, such that no two distinct edges e,f in E have a common vertex. Let \phi (l,G)\geq 0 be the number of lmatchings in G for any l\in Z_+. In many combinatorial problems it is of interest to estimate from above and below the number \phi (l,G). In the monomerdimer models in statistical mechanics, as the integer lattice Z^d, or the Bethe lattice, i.e. an infinite kregular tree, one needs to estimate the number of lmatching in bipartite regular graphs. Moreover, one needs to estimate the corresponding quantities h_G(p), called the monomerdimer 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
here.
Refreshments will be served at 2:45 p.m. (Math Lounge, MATX 1115).
