The Brown measure of the free multiplicative Brownian motion

Driver, Bruce K.; Hall, Brian C.; Kemp, Todd

doi:10.48550/arxiv.1903.11015

Cited by 8 publications

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“…As conjectured by Biane and proved by Kemp [24], g t is the large-N limit, in the sense of * -distribution, of a Brownian motion B N t in GL(N ; C). The paper [13] then computed the Brown measure of b t . This Brown measure is supported in a domain Σ t introduced by Biane and has a special structure.…”

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confidence: 99%

“…It is also related to the law ν t of the free unitary Brownian motion. The results of [13] were then extended by Ho and Zhong [23] to compute the Brown measure of ub t , where u is an arbitrary unitary element freely independent of b t .…”

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confidence: 99%

“…Following [13], it is convenient to describe these Brownian motions by one real parameter s and one complex parameter τ, where s controls the overall rate of diffusion rate, while the real and imaginary parts of τ control the diffusion rate in the unitary direction and the correlations between the unitary and positive diffusions. The complex parameter τ is nonzero and satisfies |τ − s| ≤ s. See Section 2.1 for details of the parametrization.…”

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confidence: 99%

“…where the answer will depend on the choice of the unitary initial condition u 0 in (1.4). The case τ = s corresponds to the Brown measure computed in [13,23].…”

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confidence: 99%

“…The map Γ τ1,τ2 s is illustrated in Figure 4. The paper [13] showed that the pushforward of the Brown measure µ s,s of the "standard" free multiplicative Brownian motion b s by a certain map Φ s is equal to the law of the free unitary Brownian motion u s . This result was extended in [23] to relate the Brown measure of ub s to the law of uu s , where u is an arbitrary unitary element freely independent of b s and of u s .…”

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Brown measure support and the free multiplicative Brownian motion

Hall

Kemp

2019

Advances in Mathematics

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The free multiplicative Brownian motion bt is the large-N limit of Brownian motion B N t on the general linear group GL(N ; C). We prove that the Brown measure for bt-which is an analog of the empirical eigenvalue distribution for matrices-is supported on the closure of a certain domain Σt in the plane. The domain Σt was introduced by Biane in the context of the large-N limit of the Segal-Bargmann transform associated to GL(N ; C).We also consider a two-parameter version, bs,t: the large-N limit of a related family of diffusion processes on GL(N ; C) introduced by the second author. We show that the Brown measure of bs,t is supported on the closure of a certain planar domain Σs,t, generalizing Σt, introduced by Ho.In the process, we introduce a new family of spectral domains related to any operator in a tracial von Neumann algebra: the L p n -spectrum for n ∈ N and p ≥ 1, a subset of the ordinary spectrum defined relative to potentially-unbounded inverses. We show that, in general, the support of the Brown measure of an operator is contained in its L 2 2 -spectrum. 1 arXiv:1810.00153v4 [math.FA]

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“…where the answer will depend on the choice of the unitary initial condition u 0 in (1.4). The case τ = s corresponds to the Brown measure computed in [13,23].…”

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confidence: 99%

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Brown measure support and the free multiplicative Brownian motion

Hall

Kemp

2019

Advances in Mathematics

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Brown Measures of Free Circular and Multiplicative Brownian Motions with Self-Adjoint and Unitary Initial Conditions

Ho¹,

Zhong²

2019

Preprint

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Given a selfadjoint random variable x 0 and a unitary random variable u, different from Haar unitary, free from the free circular Brownian motion c t and the free multiplicative Brownian motion b t , we use the Hamilton-Jacobi method to compute the Brown measures of free circular Brownian motion x 0 + c t and the free multiplicative Brownian motion ub t with probabilistic initial point, extending the recent work [13] by Driver-Hall-Kemp. We find that the supports of the Brown measures are related to the subordination functions of the free additive and multiplicative convolutions. The density of the Brown measure of x 0 + c t is constant along the vertical direction in the additive case, and the density of the Brown measure of ub t has the form, in polar coordinates, of (1/r 2 )w t (θ). The densities of the Brown measures are closely related to the laws of x 0 + s t and uu t , where s t is the free semicircular Brownian motion and u t is the free unitary Brownian motion, both freely independent of x 0 and u.

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PDE methods in random matrix theory

Hall

2019

Preprint

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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. Contents1. Random matrices 1.1. The Gaussian unitary ensemble 1.2. The Ginibre ensemble 1.3. The Ginibre Brownian motion 2. Large-N limits in random matrix theory 2.1. Limit in * -distribution 2.2. Tracial von Neumann algebras 2.3. Free independence 2.4. The circular Brownian motion 3. Brown measure 3.1. The goal 3.2. A motivating computation 3.3. Definition and basic properties 3.4. Brown measure in random matrix theory 3.5. The case of the circular Brownian motion 4. PDE for the circular law 4.1. The finite-N equation 4.2. A derivation using free stochastic calculus 5. Solving the equation 5.1. The Hamilton-Jacobi method 5.2. Solving the equations 6. Letting x tend to zero 6.1. Letting x tend to zero: outside the disk 6.2. Letting x tend to zero: inside the disk 6.3. On the boundary 7. The case of the free multiplicative Brownian motion 7.1. Additive and multiplicative models Supported in part by a Simons Foundation Collaboration Grant for Mathematicians.

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The Brown measure of the free multiplicative Brownian motion

Cited by 8 publications

References 36 publications

Brown measure support and the free multiplicative Brownian motion

Brown measure support and the free multiplicative Brownian motion

Brown Measures of Free Circular and Multiplicative Brownian Motions with Self-Adjoint and Unitary Initial Conditions

PDE methods in random matrix theory

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