Logarithmic terms in the spectral statistics of band random matrices

1993 ◽  
Vol 26 (15) ◽  
pp. 3845-3852 ◽  
Author(s):  
E Caurier ◽  
A Ramani ◽  
B Grammaticos
Keyword(s):  
Random Matrices ◽  
Spectral Statistics

scholarly journals Asymptotic Linear Spectral Statistics for Spiked Hermitian Random Matrices

2015 ◽  
Vol 160 (1) ◽  
pp. 120-150 ◽  
Author(s):  
Damien Passemier ◽  
Matthew R. McKay ◽  
Yang Chen
Keyword(s):  
Random Matrices ◽  
Spectral Statistics

scholarly journals Local spectral statistics of the addition of random matrices

2019 ◽  
Vol 175 (1-2) ◽  
pp. 579-654 ◽  
Author(s):  
Ziliang Che ◽  
Benjamin Landon
Keyword(s):  
Random Matrices ◽  
Spectral Statistics

scholarly journals Local laws for non-Hermitian random matrices and their products

2019 ◽  
Vol 09 (04) ◽  
pp. 2150004
Author(s):  
Friedrich Götze ◽  
Alexey Naumov ◽  
Alexander Tikhomirov
Keyword(s):  
Random Matrices ◽  
Scale Up ◽  
Random Variable ◽  
Limiting Distribution ◽  
Local Version ◽  
Logarithmic Factor ◽  
Spectral Statistics ◽  
Local Law ◽  
Further Development ◽  
Independent Identically Distributed

We consider products of independent [Formula: see text] non-Hermitian random matrices [Formula: see text]. Assume that their entries, [Formula: see text], are independent identically distributed random variables with zero mean, unit variance. Götze and Tikhomirov [On the asymptotic spectrum of products of independent random matrices, preprint (2010), arXiv:1012.2710] and O’Rourke and Sochnikov [Products of independent non-Hermitian random matrices, Electron. J. Probab. 16 (2011) 2219–2245] proved that under these assumptions the empirical spectral distribution (ESD) of [Formula: see text] converges to the limiting distribution which coincides with the distribution of the [Formula: see text]th power of random variable uniformly distributed in the unit circle. In this paper, we provide a local version of this result. More precisely, assuming additionally that [Formula: see text] for some [Formula: see text], we prove that ESD of [Formula: see text] converges to the limiting distribution on the optimal scale up to [Formula: see text] (up to some logarithmic factor). Our results generalize the recent results of Bourgade et al. [Local circular law for random matrices, Probab. Theory Related Fields 159 (2014) 545–595], Tao and Vu [Random matrices: Universality of local spectral statistics of non-Hermitian matrices, Ann. Probab. 43 (2015) 782–874] and Nemish [Local law for the product of independent non-hermitian random matrices with independent entries, Electron. J. Probab. 22 (2017) 1–35]. We also give further development of Stein’s type approach to estimate the Stieltjes transform of ESD.

scholarly journals Universality of local spectral statistics of products of random matrices

2020 ◽  
Vol 102 (5) ◽  
Author(s):  
Gernot Akemann ◽  
Zdzislaw Burda ◽  
Mario Kieburg
Keyword(s):  
Random Matrices ◽  
Products Of Random Matrices ◽  
Spectral Statistics

scholarly journals Second-Order Moment Convergence Rates for Spectral Statistics of Random Matrices

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Junshan Xie
Keyword(s):  
Random Matrices ◽  
Complete Convergence ◽  
Convergence Rates ◽  
Covariance Matrices ◽  
Second Order ◽  
Order Moment ◽  
Precise Asymptotics ◽  
Moment Convergence ◽  
Sample Covariance ◽  
Spectral Statistics

This paper considers the precise asymptotics of the spectral statistics of random matrices. Following the ideas of Gut and Spătaru (2000) and Liu and Lin (2006) on the precise asymptotics of i.i.d. random variables in the context of the complete convergence and the second-order moment convergence, respectively, we will establish the precise second-order moment convergence rates of a type of series constructed by the spectral statistics of Wigner matrices or sample covariance matrices.

scholarly journals Barrier billiard and random matrices

2021 ◽  
Author(s):  
Eugene B Bogomolny
Keyword(s):  
Quantum Mechanics ◽  
Barrier Height ◽  
Random Matrices ◽  
Low Complexity ◽  
Integrable Models ◽  
Independent Interest ◽  
Unitary Matrix ◽  
Quantum Properties ◽  
Spectral Statistics ◽  
Hopf Method

Abstract The barrier billiard is the simplest example of pseudo-integrable models with interesting and intricate classical and quantum properties. Using the Wiener-Hopf method it is demonstrated that quantum mechanics of a rectangular billiard with a barrier in the centre can be reduced to the investigation of a certain unitary matrix. Under heuristic assumptions this matrix is substituted by a special low-complexity random unitary matrix of independent interest. The main results of the paper are (i) spectral statistics of such billiards is insensitive to the barrier height and (ii) it is well described by the semi-Poisson distributions.

Sparse random matrices: spectral edge and statistics of rooted trees

2001 ◽  
Vol 33 (1) ◽  
pp. 124-140 ◽  
Author(s):  
A. Khorunzhy
Keyword(s):  
Graph Theory ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Random Matrices ◽  
Random Graphs ◽  
Rooted Trees ◽  
Theory Method ◽  
Sparse Random Matrices ◽  
Spectral Edge ◽  
Spectral Statistics

Following Füredi and Komlós, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply this method to sparse N × N random matrices AN,p that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm ∥AN,p∥ in the limit 1 ≪ p ≪ N. We show that the value pc = log N is the critical one for lim ∥AN,p/√p∥ to be bounded or not. We discuss relations of this result with the Erdős–Rényi limit theorem and properties of large random graphs. In the proof, the principal issue is that the averaged vertex degree of plane rooted trees of k edges remains bounded even when k → ∞. This observation implies fairly precise estimates for the moments of AN,p. They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectral statistics at the border of the limiting spectra of large random matrices.

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