scholarly journals Mutually avoiding paths in random media and largest eigenvalues of random matrices

2017 ◽  
Vol 95 (3) ◽  
Author(s):  
Andrea De Luca ◽  
Pierre Le Doussal
Keyword(s):  
Random Matrices ◽  
Random Media ◽  
Largest Eigenvalues ◽  
Eigenvalues Of Random Matrices

scholarly journals Small-Deviation Inequalities for Sums of Random Matrices

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 638
Author(s):  
Xianjie Gao ◽  
Chao Zhang ◽  
Hongwei Zhang
Keyword(s):  
Random Matrices ◽  
Large Deviation ◽  
Small Deviation ◽  
High Dimensional ◽  
Main Research ◽  
Infinite Dimensional ◽  
The Matrix ◽  
Largest Eigenvalues ◽  
Deviation Inequalities ◽  
Eigenvalues Of Random Matrices

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.

The statistics of the eigenvalues of random matrices

1966 ◽  
Vol 78 (3) ◽  
pp. 553-556 ◽  
Author(s):  
F. Cristofori ◽  
P.G. Sona ◽  
F. Tonolini
Keyword(s):  
Random Matrices ◽  
Eigenvalues Of Random Matrices

The RESPECT method. Simple proof of finding estimates from below for minimal singular eigenvalues of random matrices whose entries have zero means and bounded variances

2019 ◽  
Vol 27 (3) ◽  
pp. 167-175
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Lower Bounds ◽  
Simple Proof ◽  
New Method ◽  
The Matrix ◽  
Eigenvalues Of Random Matrices

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.

On the eigenvalues of random matrices

1994 ◽  
Vol 31 (A) ◽  
pp. 49-62 ◽  
Author(s):  
Persi Diaconis ◽  
Mehrdad Shahshahani
Keyword(s):  
Random Matrices ◽  
Haar Measure ◽  
Random Matrix ◽  
Unitary Group ◽  
Symmetric Groups ◽  
Random Variable ◽  
Normal Random Variable ◽  
Eigenvalues Of Random Matrices

Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.

scholarly journals Free probability for purely discrete eigenvalues of random matrices

2018 ◽  
Vol 70 (3) ◽  
pp. 1111-1150 ◽  
Author(s):  
Benoit COLLINS ◽  
Takahiro HASEBE ◽  
Noriyoshi SAKUMA
Keyword(s):  
Random Matrices ◽  
Free Probability ◽  
Eigenvalues Of Random Matrices
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