From the first rigorous proof of the Circular Law in 1984 to the Circular Law for block random matrices under the generalized Lindeberg condition

2018 ◽  
Vol 26 (2) ◽  
pp. 89-116 ◽  
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Stochastic Matrix ◽  
Rigorous Proof ◽  
Circular Law ◽  
Lindeberg Condition

Abstract The Circular Law under Lindeberg’s condition for the independent blocks of random matrices having zero expectations and double stochastic matrix of covariances of their array is proven.

𝑉-law for random block matrices under the generalized Lindeberg condition

2019 ◽  
Vol 27 (3) ◽  
pp. 177-198
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Stochastic Matrix ◽  
Block Matrices ◽  
Lindeberg Condition ◽  
Random Block ◽  
Random Block Matrices

Abstract The V-law under generalized Lindeberg condition for the independent blocks of random matrices having double stochastic matrix of covariances and different expectations of their array is proven.

VICTORIA transform, RESPECT and REFORM methods for the proof of the G-Elliptic Law under G-Lindeberg condition and twice stochastic condition for the variances and covariances of the entries of some random matrices

2020 ◽  
Vol 28 (2) ◽  
pp. 131-162
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Lindeberg Condition

AbstractThe G-Elliptic law under the G-Lindeberg condition for the independent pairs of the entries of a random matrix is proven.

V-density for eigenvalues of random block matrices with independent blocks whose entries have different variances and expectations

2019 ◽  
Vol 27 (3) ◽  
pp. 161-165
Author(s):  
Vyacheslav L. Girko ◽  
L. D. Shevchuk
Keyword(s):  
Random Matrices ◽  
Block Matrices ◽  
Lindeberg Condition ◽  
Random Block ◽  
Random Block Matrices

Abstract V-density under Lindeberg condition for the independent blocks of random matrices having different variances and expectations is found.

scholarly journals RANDOM MATRICES: THE CIRCULAR LAW

2008 ◽  
Vol 10 (02) ◽  
pp. 261-307 ◽  
Author(s):  
TERENCE TAO ◽  
VAN VU
Keyword(s):  
Strong Convergence ◽  
Random Matrices ◽  
Random Variable ◽  
Singular Value ◽  
Additive Combinatorics ◽  
Sparse Random Matrices ◽  
Circular Law ◽  
Bounded Variance ◽  
Complex Random Variable ◽  
Least Singular Value

Let x be a complex random variable with mean zero and bounded variance σ2. Let Nn be a random matrix of order n with entries being i.i.d. copies of x. Let λ1, …, λn be the eigenvalues of [Formula: see text]. Define the empirical spectral distributionμn of Nn by the formula [Formula: see text] The following well-known conjecture has been open since the 1950's: Circular Law Conjecture: μn converges to the uniform distribution μ∞ over the unit disk as n tends to infinity. We prove this conjecture, with strong convergence, under the slightly stronger assumption that the (2 + η)th-moment of x is bounded, for any η > 0. Our method builds and improves upon earlier work of Girko, Bai, Götze–Tikhomirov, and Pan–Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

scholarly journals Circular law for random matrices with unconditional log-concave distribution

2015 ◽  
Vol 17 (04) ◽  
pp. 1550020 ◽  
Author(s):  
Radosław Adamczak ◽  
Djalil Chafaï
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Real Matrix ◽  
Random Real ◽  
Empirical Spectral Distribution ◽  
Circular Law ◽  
Ginibre Ensemble ◽  
Mean And Variance ◽  
Uniform Law ◽  
Special Case

We explore the validity of the circular law for random matrices with non-i.i.d. entries. Let M be an n × n random real matrix obeying, as a real random vector, a log-concave isotropic (up to normalization) unconditional law, with mean squared norm equal to n. The entries are uncorrelated and obey a symmetric law of zero mean and variance 1/n. This model allows some dependence and non-equidistribution among the entries, while keeping the special case of i.i.d. standard Gaussian entries, known as the real Ginibre Ensemble. Our main result states that as the dimension n goes to infinity, the empirical spectral distribution of M tends to the uniform law on the unit disc of the complex plane.

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