PDE Methods in Random Matrix Theory

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. Random MatricesRandom matrix theory consists of choosing an N × N matrix at random and looking at natural properties of that matrix, notably its eigenvalues. Typically, interesting results are obtained only for large random matrices, that is, in the limit as N tends to infinity. The subject began with the work of Wigner [43], who was studying energy levels in large atomic nuclei. The subject took on new life with the discovery that the eigenvalues of certain types of large random matrices resemble the energy levels of quantum chaotic systems-that is, quantum mechanical systems for which the underlying classical system is chaotic. (See, e.g.,[20] or [39].) There is also a fascinating conjectural agreement, due to Montgomery [35], between the statistical behavior of zeros of the Riemann zeta function and the eigenvalues of random matrices. See also [30] or [6].We will review briefly some standard results in the subject, which may be found in textbooks such as those by Tao [40] or Mehta [33].

The Brown measure of the sum of a self-adjoint element and an elliptic element

“…We combine this method with techniques in free probability to determine the Brown measure of y 0 + σs− t 2 + iσ t 2 in terms of y 0 , and s and t. The results in [21] used a PDE method introduced in the work of Driver, Hall and Kemp [12]; this method has been used in subsequent work by other authors [10,21,25]. See also the expository article [19] by Hall for an introduction to the PDE method.…”

Brown measure support and the free multiplicative Brownian motion

“…The proofs use a variant of the PDE method introduced in [13], which studied the "standard" free multiplicative Brownian motion with trivial initial condition. (See also [19] for a gentle introduction to the PDE method.) In the notation of the present paper, [13] corresponds to taking τ = s and u 0 = 1.…”