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Random Matrices and Partitions

Ninness, Richard
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Thesis/Dissertation
Date
2024-05
Advisor
Rider, Brian (Brian C.)
Committee member
Yilmaz, Atilla
Dolgushev, Vasily
Group
Department
Mathematics
Subject
Mathematics
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http://hdl.handle.net/20.500.12613/10324
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DOI
http://dx.doi.org/10.34944/dspace/10286
Abstract
We begin by defining an integrable hierarchy called the KP hierarchy in terms of psuedodifferential operators. Unitary-invariant matrix ensembles provide examples of solutions to the KP hierarchy, another class of solutions to the KP hierarchy is given by Riemann surfaces with a line bundle and marked point. The latter solutions are described by the famous Krichever correspondence. At this point we also sketch how to solve Riemann Hilbert Problems which describe the asymptotics of orthogonal polynomials with varying exponential weight. Here we point out a similarity between the solutions of these Riemann Hilbert Problems and the Baker functions of the KP hierarchy. We then turn our focus to models of random three-dimensional partitions, in order to deal with these random shapes the Schur process is introduced and its correlation functions are computed. This gives us a way to find asymptotics for plane partitions and pyramid partitions in the case where all q parameters are going to 1 together. Finally, in order to find more precise asymptotics for pyramid partitions in which one parameter is fixed we connect pyramid partitions to random matrices. Specifically, the Stieltjes-Wigert model is defined and it is shown how the double scaling limits of moments of the Stieltjes-Wigert model completely determines the frozen boundary of pyramid partitions.
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