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PDE methods in random matrix theory

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 Added by Brian C. Hall
 Publication date 2019
  fields Physics
and research's language is English
 Authors Brian C. Hall




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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.



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The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -- coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.
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In this paper we consider an asymptotic question in the theory of the Gaussian Unitary Ensemble of random matrices. In the bulk scaling limit, the probability that there are no eigenvalues in the interval (0,2s) is given by P_s=det(I-K_s), where K_s is the trace-class operator with kernel K_s(x,y)={sin(x-y)}/{pi(x-y)} acting on L^2(0,2s). We are interested particularly in the behavior of P_s as s tends to infinity...
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