Abstract

In this thesis we consider two classical combinatorial problems arising in statistical mechanics: counting matchings and self-avoiding walks in lattice graphs. The first problem arises in the study of the thermodynamical properties of monomers and dimers (diatomic molecules) in crystals. Fisher, Kasteleyn and Temperley discovered an elegant technique to exactly count the number of perfect matchings in two dimensional lattices, but it is not applicable for matchings of arbitrary size, or in higher dimensional lattices. We present the first efficient approximation algorithm for computing the number of matchings of any size in any periodic lattice in arbitrary dimension. The algorithm is based on Monte Carlo simulation of a suitable Markov chain and has rigorously derived performance guarantees that do not rely on any assumptions. In addition, we show that these results generalize to counting matchings in any graph which is the Cayley graph of a finite group. The second problem is countin...

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