The Orthogonal Weingarten Calculus and Free Probability
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Authors
Gawlak, Dylan
Date
2024-09-05
Type
thesis
Language
eng
Keyword
Free Probability , Random Matrix Theory , Combinatorics , Mathematics , Orthogonal
Alternative Title
Abstract
We provide a self-contained derivation of the Weingarten calculus for compact matrix groups. We also provide a self-contained introduction to the combinatorics of the symmetric group needed for computations in free probability. We introduce partitioned pairings, an analogue of partitioned permutations. Using partitioned pairings, we will derive expressions for the leading and subleading terms of the Weingarten function for the orthogonal group. Using the combinatorial theory that we have developed, as well as what is called second-order real freeness, we will determine the first and second-order limiting distributions of the random matrix ensemble Xm = O1+Ot1+· · +·Om+Otm where {Oi} are independent Haar-distributed orthogonal matrices.
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Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada
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Copying and Preserving Your Thesis
This 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.
ProQuest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This 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.