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Title: Rectangular finite free probability theory
Authors: Gribinski, Aurelien Xavier
Advisors: Marcus, Adam W
Contributors: Applied and Computational Mathematics Department
Subjects: Mathematics
Issue Date: 2022
Publisher: Princeton, NJ : Princeton University
Abstract: We study a new type of polynomial convolution that serves as the foundation for building what we call rectangular finite free probability theory, generalizing the square finite free probability theory of Marcus, Spielman and Srivastava. We relate this operation to large rectangular random matrices and explain how it acts on singular values of rectangular matrices in a canonical way. Furthermore, we obtain nontrivial inequalities on roots of polynomials and build some appropriate tools, e.g. the analogue of the classical $R$-transforms. These developments are inspired by well-known results and concepts from probability theory. We also show that classical orthogonal polynomials such as Gegenbauer or Laguerre polynomials naturally arise through this convolution. Consequently, we deduce new nontrivial properties about the positions of the roots of these polynomials. As an application, we give an elegant proof of the existence of biregular bipartite Ramanujan graphs.
Alternate format: The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog:
Type of Material: Academic dissertations (Ph.D.)
Language: en
Appears in Collections:Applied and Computational Mathematics

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