Return to search

Topics in Combinatorics and Random Matrix Theory

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. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2009-09-27 12:27:21.479

Identiferoai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OKQ.1974/5235
Date27 September 2009
CreatorsNovak, JONATHAN
ContributorsQueen's University (Kingston, Ont.). Theses (Queen's University (Kingston, Ont.))
Source SetsLibrary and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada
LanguageEnglish, English
Detected LanguageEnglish
TypeThesis
Format1024235 bytes, application/pdf
RightsThis publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.
RelationCanadian theses

Page generated in 0.002 seconds