scholarly journals Singularity of sparse random matrices: simple proofs

2021 ◽  
pp. 1-8
Author(s):  
Asaf Ferber ◽  
Matthew Kwan ◽  
Lisa Sauermann
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Bernoulli Model ◽  
Sparse Random Matrices ◽  
Combinatorial Model

Abstract Consider a random $n\times n$ zero-one matrix with ‘sparsity’ p, sampled according to one of the following two models: either every entry is independently taken to be one with probability p (the ‘Bernoulli’ model) or each row is independently uniformly sampled from the set of all length-n zero-one vectors with exactly pn ones (the ‘combinatorial’ model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$ , then our random matrix is nonsingular with probability $1-o(1)$ . In the Bernoulli model, this fact was already well known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person.

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

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.

scholarly journals Local law and Tracy–Widom limit for sparse random matrices

2017 ◽  
Vol 171 (1-2) ◽  
pp. 543-616 ◽  
Author(s):  
Ji Oon Lee ◽  
Kevin Schnelli
Keyword(s):  
Random Matrices ◽  
Sparse Random Matrices ◽  
Local Law

scholarly journals On the limiting spectral density of random matrices filled with stochastic processes

2019 ◽  
Vol 27 (2) ◽  
pp. 89-105 ◽  
Author(s):  
Matthias Löwe ◽  
Kristina Schubert
Keyword(s):  
Stochastic Process ◽  
Spectral Density ◽  
Random Matrices ◽  
Markov Processes ◽  
Random Matrix ◽  
Matrix Theory ◽  
Gibbs Measures ◽  
State Spaces ◽  
The Matrix ◽  
Finite State

Abstract We discuss the limiting spectral density of real symmetric random matrices. In contrast to standard random matrix theory, the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well-known semi-circle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.

scholarly journals The Expected Norm of Random Matrices

2000 ◽  
Vol 9 (2) ◽  
pp. 149-166 ◽  
Author(s):  
YOAV SEGINER
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Operator Norm ◽  
Euclidean Norm ◽  
Constant Independent ◽  
Multiplicative Constant ◽  
The Matrix

We compare the Euclidean operator norm of a random matrix with the Euclidean norm of its rows and columns. In the first part of this paper, we show that if A is a random matrix with i.i.d. zero mean entries, then E∥A∥h [les ] Kh (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h), where K is a constant which does not depend on the dimensions or distribution of A (h, however, does depend on the dimensions). In the second part we drop the assumption that the entries of A are i.i.d. We therefore consider the Euclidean operator norm of a random matrix, A, obtained from a (non-random) matrix by randomizing the signs of the matrix's entries. We show that in this case, the best inequality possible (up to a multiplicative constant) is E∥A∥h [les ] (c log1/4 min {m, n})h (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h) (m, n the dimensions of the matrix and c a constant independent of m, n).

scholarly journals FLUCTUATIONS OF MATRIX ENTRIES OF ANALYTIC FUNCTIONS OF NON-HERMITIAN RANDOM MATRICES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250008
Author(s):  
SEAN O'ROURKE
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Analytic Functions ◽  
Random Variables

Consider an n × n non-Hermitian random matrix Mn whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of f(Mn) as n tends to infinity, where f is analytic on an appropriate domain. This extends the results in [19, 20, 23] from symmetric random matrices to the non-Hermitian case.

A numerical study of sparse random matrices

1992 ◽  
Vol 69 (1-2) ◽  
pp. 361-383 ◽  
Author(s):  
S. N. Evangelou
Keyword(s):  
Random Matrices ◽  
Numerical Study ◽  
Sparse Random Matrices
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