Limiting Spectral Radii of Circular Unitary Matrices Under Light Truncation
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Limiting Spectral Radii of Circular Unitary Matrices Under Light Truncation Yu Miao1 · Yongcheng Qi2 Received: 21 February 2020 / Revised: 23 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Consider a truncated circular unitary matrix which is a pn by pn submatrix of an n by n circular unitary matrix after deleting the last n − pn columns and rows. Jiang and Qi (J Theor Probab 30:326–364, 2017) and Gui and Qi (J Math Anal Appl 458:536–554, 2018) 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). pn /n is bounded away from 0 and 1; (2). pn → ∞ and pn /n → 0 as n → ∞; (3). (n − pn )/n → 0 and (n − pn )/(log n)3 → ∞ as n → ∞; (4). n − pn → ∞ and (n − pn )/ log n → 0 as n → ∞; and (5). n − pn = 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 − pn 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 (2018). Keywords Spectral radius · Eigenvalue · Limiting distribution · Extreme value · Circular unitary matrix Mathematics Subject Classification (2010) 60F99 · 60G55 · 60G70
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Yongcheng Qi [email protected] Yu Miao [email protected]
1
College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, Henan Province, China
2
Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812, USA
123
Journal of Theoretical Probability
1 Introduction The study of large random matrices can date back to nearly a century ago, and one example is Wishart’s [33] work on statistical properties for large covariance matrices. The theory of random matrices has been rapidly developed in last few decades and has found applications in heavy-nuclei atoms [32], number theory [23], quantum mechanics [22], condensed matter physics [11], wireless communications [7], to just mention a few. Statistical properties of large random matrices including their empirical spectral distributions and spectral radii (the largest eigenvalues) are of particular interest in the study. For the three Hermitian matrices including Gaussian orthogonal ensemble, Gaussian unitary ensemble and Gaussian symplectic ensemble, Tracy and Widom [29,30] show that their spectral radii converge in distribution to Tracy–Widom laws. For more consequent applications of Tracy–Widom laws, see, e.g., Baik et al. [3], Tracy and Widom [31], Johansson [19], Johnstone [20,21], and Jiang [15]. For a non-Hermitian matrix, the largest absolute value of its eigenvalues is referred to as the spectral radius. The spectral
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