The Laguerre process and generalized Hartman–Watson law

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doi:10.3150/07-bej6048

Cited by 22 publications

(41 citation statements)

References 16 publications

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“…Other processes occur in the study of rescaled versions of the eigenvalues processes of other random matricesr. In particular, the Laguerre process arises as the scaling limit of the low-lying eigenvalues of Wishart matrices, see [Bru91], [KoO01] and [Dem07], and has the interpretation of Bessel processes conditioned not to intersect.…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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“…Then, as proven in [40], [24] the solution of the system of SDEs (2) can be realized as N independent copies of G (β) -diffusions conditioned to never intersect via a Doob htransform. The corresponding transition kernel is then given by the Doob h-transformed Karlin-McGregor determinant [36] defined by,…”

Section: The Laguerre Process and Its Eigenvaluesmentioning

confidence: 95%

“…The analogous process on real symmetric matrices was first considered by Bru in [18] under the name of Wishart process. The Hermitian case that we will be concerned with was then introduced in [40] and further studied in [24]. Let (W t ; t ≥ 0) be an N × N complex Brownian matrix.…”

Section: The Laguerre Process and Its Eigenvaluesmentioning

confidence: 99%

“…Let W N,(β) (t); t ≥ 0 be the corresponding Markov semigroup. By [24], see also Section 3 of [18], or for general affine processes (the Laguerre/Wishart is a special case) by Section 3 of [22] (again this is for real symmetric matrices but the same arguments give the result in the Hermitian case) this semigroup is actually Feller.…”

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confidence: 96%

“…The proof is completely analogous to the ones of Proposition 1.2 and Proposition 1.3. One now uses the connection to the stationary Laguerre process (Y t ; t ≥ 0), solution of the matrix SDE: [24], [18], [22]. As before, we have:…”

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confidence: 99%

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On a gateway between the Laguerre process and dynamics on partitions

Assiotis¹

2019

ALEA

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Probability measures and stochastic dynamics on matrices and on partitions are related by standard, albeit technical, discrete to continuous scaling limits. In this paper we provide exact relations, that go in both directions, between the eigenvalues of the Laguerre process and certain distinguished dynamics on partitions. This is done by generalizing to the multidimensional setting recent results of Miclo and Patie on linear one-dimensional diffusions and birth and death chains. As a corollary, we obtain an exact relation between the Laguerre and Meixner ensembles. Finally, we explain the deep connections with the Young bouquet and the z-measures on partitions.

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Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture

Banks

Kulkarni

Mukherjee

et al. 2021

Comm Pure Appl Math

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A matrix is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E. B. Davies: for each , every matrix is at least ‐close to one whose eigenvectors have condition number at worst , for some depending only on n. We further show that the dependence on δ cannot be improved to for any constant .Our proof uses tools from random matrix theory to show that the pseudospectrum of A can be regularized with the addition of a complex Gaussian perturbation. Along the way, we explain how a variant of a theorem of Śniady implies a conjecture of Sankar, Spielman, and Teng on the optimal constant for smoothed analysis of condition numbers. © 2021 Wiley Periodicals, Inc.

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The Laguerre process and generalized Hartman–Watson law

Cited by 22 publications

References 16 publications

An Introduction to Random Matrices

An Introduction to Random Matrices

On a gateway between the Laguerre process and dynamics on partitions

Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture

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