# Perfect matchings: Modified Aztec diamonds, covering graphs andn-matchings

Fall 1997

Dissertation

Mathematics

## Degree Name

Doctor of Philosophy

David V Feldman

## Abstract

In the Introduction, we present the problems we are going to study and we establish the basic definitions, concepts and results that are used throughout.

We begin the first chapter with a presentation of the Aztec diamond and the behaviour of its random domino tilings. We introduce the dual-matching-problem and we explore the structure of the perfect matchings of modified Aztec diamonds. We show that some of these matchings can be extended to matchings of the dual Aztec diamond, pointing out a bijection between these types of matchings. We determine the number of perfect matchings for each of the modified graphs and the placement probabilities of the edges belonging to such a matching at a given location. We conclude with a theorem presenting the common asymptotic behaviour of the dual and the modified Aztec diamonds and we deduce a version of the Arctic Circle Theorem for these graphs.

The second part is dedicated to the study of non-ramified perfect n-matchings, their decomposition into perfect matchings and 2-matchings as well as their relations to the perfect matchings of covering graphs. For the n-covering graphs we use the permutation derived graph construction. We determine the number of liftings of a given n-matching to a matching of a branched covering graph and then of a n-covering graph, together with necessary and sufficient conditions for the existence of the lifting. In particular, for the case of 2-matchings, we obtain a uniform behaviour of liftings of cycles. First, we deduce a theorem that relates the number of perfect matchings of the branched covering graph we have introduced to the number of perfect 2-matchings of the initial graph. Then we study the 2-covering graphs, their number, we determine the number of liftings of a 2-matchings (as a power of 2) and we obtain a theorem that characterizes the 2-matchings as the average of perfect matchings of 2-covering graphs. We conclude with some considerations about the maximum, minimum and the realization of this average and methods of computing it.

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