Brown measure support and the free multiplicative Brownian motion

Hall, Brian C.; Kemp, Todd

doi:10.1016/j.aim.2019.106771

Cited by 13 publications

(16 citation statements)

References 47 publications

(231 reference statements)

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“…In the case that Z is distributed according to the Gaussian unitary ensemble, the Itô term is Itô = 1 2 I. In this case, the resulting multiplicative model may be described as Brownian motion obtained by the author with Kemp [26]; we prove that the Brown measure of b t is supported on the closure of t . Now, we have already noted that t is simply connected for t ≤ 4 but doubly connected for t > 4.…”

Section: On the Boundarymentioning

confidence: 80%

“…As we have noted, the domains t were introduced by Biane in [4]. Two subsequent works in the physics literature, the article [18] by Gudowska-Nowak, Janik, Jurkiewicz, and Nowak and the article [32] by Lohmayer, Neuberger, and Wettig then argued, using nonrigorous methods, that the eigenvalues of B N t should concentrate into t for large N. The first rigorous result in this direction was We note, however, that none of the papers [18,32], or [26] says anything about the distribution of μ b t within t ; they are only concerned with identifying the region t . The actual computation of μ b t (not just its support) was done in [10].…”

Section: The Support Of the Brown Measure Of B Tmentioning

confidence: 98%

“…The methods of [26] explain this apparent coincidence, using the "free Hall transform" G t of Biane [4]. Biane constructed this transform using methods of free probability as an infinite-dimensional analog of the Segal-Bargmann transform for U(N), which was developed by the author in [21].…”

Section: On the Boundarymentioning

confidence: 99%

“…Recall from Section 7.2 that the distribution of the free unitary Brownian motion is Biane's measure ν t on the unit circle, the support of which is described in Theorem 19. A key ingredient in [26] is the function f t given by…”

Section: On the Boundarymentioning

confidence: 99%

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PDE Methods in Random Matrix Theory

Hall

2020

Springer Optimization and Its Applications

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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. 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].

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Section: On the Boundarymentioning

confidence: 80%

Section: The Support Of the Brown Measure Of B Tmentioning

confidence: 98%

Section: On the Boundarymentioning

confidence: 99%

Section: On the Boundarymentioning

confidence: 99%

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PDE Methods in Random Matrix Theory

Hall

2020

Springer Optimization and Its Applications

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“…In my talk, I will discuss results with Bruce Driver and Todd Kemp concerning the distribution of the eigenvalues of in the large-limit. Our first result [2] is that the eigenvalues cluster as → ∞ into a certain domain Σ in the plane identified by Biane [1]. We then have work in progress indicating a remarkably simple structure to the limiting distribution of eigenvalues within this domain.…”

Section: Brownian Motion In the General Linear Groupmentioning

confidence: 81%

Eigenvalues of Random Matrices in the General Linear Group in the Large-$N$ Limit

Hall¹

2019

Notices Amer. Math. Soc.

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The Brown measure of the free multiplicative Brownian motion

Driver

Hall

Kemp

2022

Probab. Theory Relat. Fields

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The free multiplicative Brownian motion $$b_{t}$$ b t is the large-N limit of the Brownian motion on $$\mathsf {GL}(N;\mathbb {C}),$$ GL ( N ; C ) , in the sense of $$*$$ ∗ -distributions. The natural candidate for the large-N limit of the empirical distribution of eigenvalues is thus the Brown measure of $$b_{t}$$ b t . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $$\Sigma _{t}$$ Σ t that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $$W_{t}$$ W t on $$\overline{\Sigma }_{t},$$ Σ ¯ t , which is strictly positive and real analytic on $$\Sigma _{t}$$ Σ t . This density has a simple form in polar coordinates: $$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ W t ( r , θ ) = 1 r 2 w t ( θ ) , where $$w_{t}$$ w t is an analytic function determined by the geometry of the region $$\Sigma _{t}$$ Σ t . We show also that the spectral measure of free unitary Brownian motion $$u_{t}$$ u t is a “shadow” of the Brown measure of $$b_{t}$$ b t , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

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Brown measure support and the free multiplicative Brownian motion

Cited by 13 publications

References 47 publications

PDE Methods in Random Matrix Theory

PDE Methods in Random Matrix Theory

Eigenvalues of Random Matrices in the General Linear Group in the Large-$N$ Limit

The Brown measure of the free multiplicative Brownian motion

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