Related papers: Universality of the least singular value for spars…

We discuss applications of some concepts of Compressed Sensing in the recent work on invertibility of random matrices due to Rudelson and the author. We sketch an argument leading to the optimal bound N^{-1/2} on the median of the smallest…

In this paper we consider ensemble of random matrices $\X_n$ with independent identically distributed vectors $(X_{ij}, X_{ji})_{i \neq j}$ of entries. Under assumption of finite fourth moment of matrix entries it is proved that empirical…

We consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a…

Let $M_n$ denote a random symmetric $n \times n$ matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely…

Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudo-unitary group. Further, we develop a random matrix theory which is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices as…

This paper establishes a new comparison principle for the minimum eigenvalue of a sum of independent random positive-semidefinite matrices. The principle states that the minimum eigenvalue of the matrix sum is controlled by the minimum…

An exchangeable random matrix is a random matrix with distribution invariant under any permutation of the entries. For such random matrices, we show, as the dimension tends to infinity, that the empirical spectral distribution tends to the…

Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value of A is of order n^{-1/2} with high probability. The lower estimate of this type…

We obtain a tail bound for the least non-zero singular value of $A-z$ when $A$ is a random matrix and $z$ is an eigenvalue of $A$ in a neighbourhood of a given point $z_0$ in the bulk of the spectrum. The argument relies on a resolvent…

We provide a polynomial lower bound on the minimum singular value of an $m\times m$ random matrix $M$ with jointly Gaussian entries, under a polynomial bound on the matrix norm and a global small-ball probability bound $$\inf_{x,y\in…

Let $\{a_{ij}\}$ $(1\le i,j<\infty)$ be i.i.d. real valued random variables with zero mean and unit variance and let an integer sequence $(N_m)_{m=1}^\infty$ satisfy $m/N_m\longrightarrow z$ for some $z\in(0,1)$. For each $m\in{\mathbb N}$…

We investigate the eigenvalue density in ensembles of large sparse Bernoulli random matrices. We demonstrate that the fraction of linear subgraphs just below the percolation threshold is about 95\% of all finite subgraphs, and the…

We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random with replacement. These matrices have a one-to-one correspondence with the…

Let $\mathbf{A}$ be an $n\times n$-matrix over $\mathbb{F}_2$ whose every entry equals $1$ with probability $d/n$ independently for a fixed $d>0$. Draw a vector $\mathbf{y}$ randomly from the column space of $\mathbf{A}$. It is a simple…

Let $A$ be a real skew-symmetric Gaussian random matrix whose upper triangular elements are independently distributed according to the standard normal distribution. We provide the distribution of the largest singular value $\sigma_1$ of…

We prove the universality of the joint distribution of an eigenvalue and the corresponding diagonal eigenvector overlap, in the bulk and at the edge, for eigenvalues of complex matrices and real eigenvalues of real matrices. As part of the…

The spectral moments of ensembles of sparse random block matrices are analytically evaluated in the limit of large order. The structure of the sparse matrix corresponds to the Erd\"os-Renyi random graph. The blocks are i.i.d. random…

Let $A = (a_{ij})$ be a square $n\times n$ matrix with i.i.d. zero mean and unit variance entries. Rudelson and Vershynin showed that the upper bound for a smallest singular value $s_n(A)$ is of order $n^{-\frac12}$ with probability close…

We investigate the level spacing distribution for three ensembles of real symmetric matrices having additional structural constraint to reduce the number of independent entries to only $ (n+1)/2 $ in contrast to the $ n(n+1)/2 $ for a real…

The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important…