Topics in Combinatorics and Random Matrix Theory
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Motivated by the longest increasing subsequence problem, we examine sundry topics at the interface of enumerative/algebraic combinatorics and random matrix theory. We begin with an expository account of the increasing subsequence problem, contextualizing it as an ``exactly solvable'' Ramsey-type problem and introducing the RSK correspondence. New proofs and generalizations of some of the key results in increasing subsequence theory are given. These include Regev's single scaling limit, Gessel's Toeplitz determinant identity, and Rains' integral representation. The double scaling limit (Baik-Deift-Johansson theorem) is briefly described, although we have no new results in that direction. Following up on the appearance of determinantal generating functions in increasing subsequence type problems, we are led to a connection between combinatorics and the ensemble of truncated random unitary matrices, which we describe in terms of Fisher's random-turns vicious walker model from statistical mechanics. We prove that the moment generating function of the trace of a truncated random unitary matrix is the grand canonical partition function for Fisher's random-turns model with reunions. Finally, we consider unitary matrix integrals of a very general type, namely the ``correlation functions'' of entries of Haar-distributed random matrices. We show that these expand perturbatively as generating functions for class multiplicities in symmetric functions of Jucys-Murphy elements, thus addressing a problem originally raised by De Wit and t'Hooft and recently resurrected by Collins. We argue that this expansion is the CUE counterpart of genus expansion.