Spectral radii of truncated circular unitary matrices

Gui, Wenhao; Qi, Yongcheng

doi:10.1016/j.jmaa.2017.09.030

Cited by 13 publications

(22 citation statements)

References 26 publications

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“…Apparently, the conditions above do not cover the case when n − p n is of order between log n and (log n) 3 . In this paper, we prove that the spectral radius converges in distribution to the Gumbel distribution as well in this case, as conjectured by Gui and Qi [13].…”

supporting

confidence: 73%

“…Two recent papers by Jiang and Qi [17] and Gui and Qi [13] study the limiting distributions of the spectral radius max 1≤j≤p |z j | for the truncated circular unitary ensemble. Jiang and Qi [17] have proved that the spectral radius max 1≤j≤p |z j | converges to the Gumbel distribution when the ratio p n /n is bounded away from 0 and 1.…”

Section: Introductionmentioning

confidence: 99%

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Limiting Spectral Radii of Circular Unitary Matrices under Light Truncation

Miao

2020

Preprint

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Consider a truncated circular unitary matrix which is a p n by p n submatrix of an n by n circular unitary matrix after deleting the last n − p n columns and rows. Jiang and Qi [17] and Gui and Qi [13] study the limiting distributions of the maximum absolute value of the eigenvalues (known as spectral radius) of the truncated matrix. Some limiting distributions for the spectral radius for the truncated circular unitary matrix have been obtained under the following conditions: (1). p n /n is bounded away from 0 and 1; (2). p n → ∞ and p n /n → 0 as n → ∞; (3). (n − p n )/n → 0 andand (5). n − p n = k ≥ 1 is a fixed integer. The spectral radius converges in distribution to the Gumbel distribution under the first four conditions and to a reversed Weibull distribution under the fifth condition. Apparently, the conditions above do not cover the case when n − p n is of order between log n and (log n) 3 . In this paper, we prove that the spectral radius converges in distribution to the Gumbel distribution as well in this case, as conjectured by Gui and Qi [13].

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supporting

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Limiting Spectral Radii of Circular Unitary Matrices under Light Truncation

Miao

2020

Preprint

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“…Recently, Lacroix-A-Chez-Toine, Grabsch, Majumdar and Schehr [LACTGMS18] showed a universal intermediate deviations regime for different Coulomb gases. The farthest particles of further models are studied in [Seo15], [GQ18] and [CLQ18]. Some other weakly confining models and strongly confining models are studied by one of the authors in [GZ18b] mostly by the techniques used in this article.…”

Section: The Coulomb Gas Modelmentioning

confidence: 99%

Extremal particles of two-dimensional Coulomb gases and random polynomials on a positive background

Butez¹,

García-Zelada²

2018

Preprint

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We study the extremal particles of the two-dimensional Coulomb gas with confinement generated by a radially symmetric positive background in the determinantal case and the zeros of the corresponding random polynomials. We show that when the background is supported on the unit disk, the point process of the particles outside of the disk converges towards a universal point process, i.e. that does not depend on the background. This limiting point process may be seen as the determinantal point process governed by the Bergman kernel on the complement of the unit disk. It has an infinite number of particles and its maximum is a heavy tailed random variable.To prove this convergence we study the case where the confinement is generated by a positive background outside of the unit disk. For this model we show that the point process of the particles inside the disk converges towards the determinantal point process governed by the Bergman kernel on the unit disk. In the case where the background is unbounded, we study the speed at which the farthest particle of the Coulomb gas converges to infinity, and we obtain similar results for the associated random polynomials.

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“…These limiting distributions are no longer the Tracy-Widom laws. Gui and Qi (2018) Let m ≥ 1 be an integer and assume X 1 , · · · , X m are m independent and identically distributed (i.i.d.) n × n random matrices.…”

Section: Introductionmentioning

confidence: 99%

Limiting distributions of spectral radii for product of matrices from the spherical ensemble

Chang

2018

Journal of Mathematical Analysis and Applications

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Consider the product of m independent n × n random matrices from the spherical ensemble for m ≥ 1. The spectral radius is defined as the maximum absolute value of the n eigenvalues of the product matrix. When m = 1, the limiting distribution for the spectral radii has been obtained by Jiang and Qi (2017). In this paper, we investigate the limiting distributions for the spectral radii in general. When m is a fixed integer, we show that the spectral radii converge weakly to distributions of functions of independent Gamma random variables. When m = m n tends to infinity as n goes to infinity, we show that the logarithmic spectral radii have a normal limit.

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Spectral radii of truncated circular unitary matrices

Abstract: Consider a truncated circular unitary matrix which

Cited by 13 publications

References 26 publications

Limiting Spectral Radii of Circular Unitary Matrices under Light Truncation

Limiting Spectral Radii of Circular Unitary Matrices under Light Truncation

Extremal particles of two-dimensional Coulomb gases and random polynomials on a positive background

Limiting distributions of spectral radii for product of matrices from the spherical ensemble

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