Regularization of Non-Normal Matrices by Gaussian Noise

Feldheim, Ohad N.; Paquette, Elliot; Zeitouni, Ofer

doi:10.1093/imrn/rnu213

Cited by 14 publications

(14 citation statements)

References 12 publications

(15 reference statements)

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“…The choice of γ in each of the three cases given in Corollary 1.8 is so that (1.11) is satisfied. Theorem 1.7 and its corollary are similar to several recent works concerning fixed matrices perturbed by random matrices [7,8,10,29,60,61,62,73]. The case when E contains independent Gaussian entires was investigated in [7,29,60,61,62].…”

Section: 14)supporting

confidence: 59%

Quantitative results for non-normal matrices subject to random and deterministic perturbations

O’Rourke¹,

Wood²

2021

Preprint

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We consider the eigenvalue distribution of a fixed matrix subject to a small perturbation. In particular, we prove quantitative comparison results which show that the logarithmic potential is stable under perturbations with small norm or low rank, provided the smallest and largest singular values are not too extreme. We also establish a quantitative version of the Tao-Vu replacement principle. As an application of our results, we study the spectral distribution of banded Toeplitz matrices subject to small random perturbations, including the case when the bandwidth grows with the dimension. For this model, we obtain a rate of convergence in Wasserstein distance of the empirical spectral measure to its limiting distribution. We give a number of examples of our methods including random multiplicative perturbations and large classes of deterministic perturbations that behave similar to random perturbations.

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Section: 14)supporting

confidence: 59%

Quantitative results for non-normal matrices subject to random and deterministic perturbations

O’Rourke¹,

Wood²

2021

Preprint

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“…In [28], Feldheim, Paquette and Zeitouni have recently studied the model (1.2) when σ decays polynomially of N and A N is a block diagonal matrix with blocks of size log N .…”

Section: Discussion and Related Resultsmentioning

confidence: 99%

Outlier Eigenvalues for Deformed I.I.D. Random Matrices

Bordenave

Capitaine

2016

Comm Pure Appl Math

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We consider a square random matrix of size N of the form A + Y where A is deterministic and Y has i.i.d. entries with variance 1/N. Under mild assumptions, as N grows the empirical distribution of the eigenvalues of A + Y converges weakly to a limit probability measure β on the complex plane. This work is devoted to the study of the outlier eigenvalues, i.e., eigenvalues in the complement of the support of β. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of Y or the Jordan decomposition of A. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function. © 2016 Wiley Periodicals, Inc.

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“…Śniady shows that if we add to X N a small amount of random Gaussian noise, then eigenvalues distribution of the perturbed matrices will converge to the Brown measure of the limiting object. (See also the papers [19] and [12], which obtain similar results by very different methods.) Thus, if the original random matrices X N are somehow "stable," adding this noise should not change the eigenvalues of X N by much, and the eigenvalues of the original and perturbed matrices should be almost the same.…”

Section: Brown Measure In Random Matrix Theorymentioning

confidence: 86%

“…In particular, unless x is normal, this integral need not be equal to τ [x k (x * ) l ]. Thus, to compute the Brown measure of a general operator x ∈ A, we actually have to work with the rather complicated definition in (11), (12), and (13).…”

Section: Definition and Basic Propertiesmentioning

confidence: 99%

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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Regularization of Non-Normal Matrices by Gaussian Noise

Cited by 14 publications

References 12 publications

Quantitative results for non-normal matrices subject to random and deterministic perturbations

Quantitative results for non-normal matrices subject to random and deterministic perturbations

Outlier Eigenvalues for Deformed I.I.D. Random Matrices

PDE Methods in Random Matrix Theory

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