scholarly journals Sparse random matrices have simple spectrum

2020 ◽  
Vol 56 (4) ◽  
pp. 2307-2328
Author(s):  
Kyle Luh ◽  
Van Vu
Keyword(s):  
Random Matrices ◽  
Simple Spectrum ◽  
Sparse Random Matrices

Communication Optimal Parallel Multiplication of Sparse Random Matrices

2013 ◽  
Author(s):  
Grey Ballard ◽  
Aydin Buluc ◽  
James Demmel ◽  
Laura Grigori ◽  
Benjamin Lipshitz ◽  
...  
Keyword(s):  
Random Matrices ◽  
Sparse Random Matrices ◽  
Parallel Multiplication

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.

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