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PDE Methods in Random Matrix Theory

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Harmonic Analysis and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 168))

Abstract

This article begins with a brief review of random matrix theory, followed by a discussion of how the large-N limit of random matrix models can be realized using operator algebras. I then explain the notion of “Brown measure,” which play the role of the eigenvalue distribution for operators in an operator algebra.

I then show how methods of partial differential equations can be used to compute Brown measures. I consider in detail the case of the circular law and then discuss more briefly the case of the free multiplicative Brownian motion, which was worked out recently by the author with Driver and Kemp.

Supported in part by a Simons Foundation Collaboration Grant for Mathematicians

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, USA

    Brian C. Hall

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  1. Brian C. Hall
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Correspondence to Brian C. Hall .

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Editors and Affiliations

  1. Institute of Mathematics, University of Zurich, Zurich, Switzerland

    Michael Th. Rassias

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Hall, B.C. (2021). PDE Methods in Random Matrix Theory. In: Rassias, M.T. (eds) Harmonic Analysis and Applications. Springer Optimization and Its Applications, vol 168. Springer, Cham. https://doi.org/10.1007/978-3-030-61887-2_5

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