eigenvalues of random matrices
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Author(s):  
Renjie Feng ◽  
Gang. Tian ◽  
Dongyi. Wei

In our previous paper [R. Feng, G. Tian and D. Wei, Spectrum of SYK model, Peking Math. J. 2 (2019) 41–70], we derived the almost sure convergence of the global density of eigenvalues of random matrices of the SYK model. In this paper, we will prove the central limit theorem for the linear statistics of eigenvalues of the SYK model and compute its variance.


2019 ◽  
Vol 27 (3) ◽  
pp. 167-175
Author(s):  
Vyacheslav L. Girko

Abstract The lower bounds for the minimal singular eigenvalue of the matrix whose entries have zero means and bounded variances are obtained. The new method is based on the G-method of perpendiculars and the RESPECT method.


Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 638
Author(s):  
Xianjie Gao ◽  
Chao Zhang ◽  
Hongwei Zhang

Random matrices have played an important role in many fields including machine learning, quantum information theory, and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they are applicable to high-dimensional and even the infinite-dimensional cases.


Author(s):  
Thomas Spencer

This article examines some of the connections between random matrix theory (RMT) and number theory, including the modelling of the value distributions of the Riemann zeta function and other L-functions as well as the statistical distribution of their zeros. Number theory has been used in RMT to address seemingly disparate questions, such as modelling mean and extreme values of the Riemann zeta function and counting points on curves. One thing in common among the applications of RMT to number theory is the L-function. The statistics of the critical zeros of these functions are believed to be related to those of the eigenvalues of random matrices. The article first considers the truth of the generalized Riemann hypothesis before discussing the values of the Riemann zeta function, the values of L-functions, and further areas of interest with respect to the connections between RMT and number theory


2018 ◽  
Vol 70 (3) ◽  
pp. 1111-1150 ◽  
Author(s):  
Benoit COLLINS ◽  
Takahiro HASEBE ◽  
Noriyoshi SAKUMA

2018 ◽  
Vol 26 (2) ◽  
pp. 117-123 ◽  
Author(s):  
Vyacheslav L. Girko

Abstract The lower bounds for the minimal singular eigenvalue of the matrix are obtained under the G-Lindeberg condition and the G-double stochastic condition for the variances of the matrix entries. The new method is based on the G-method of perpendiculars, the REFORM method, the martingale method, and the theory of canonical spectral equations.


2017 ◽  
Vol 167 (2) ◽  
pp. 234-259 ◽  
Author(s):  
Aurélien Grabsch ◽  
Satya N. Majumdar ◽  
Christophe Texier

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