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A fooling-set matrix has nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. '96 proved that the rank of such a matrix is at least $\sqrt n$. It is known that the bound is tight (up to…

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors $\pm X_1,...,\pm X_N\in\R^n$, ($N\ge n$). We introduce a class of random sampling matrices and show that they…

Consider random matrices $A$, of dimension $m\times (m+n)$, drawn from an ensemble with probability density $f(\rmtr AA^\dagger)$, with $f(x)$ a given appropriate function. Break $A = (B,X)$ into an $m\times m$ block $B$ and the…

We explore the validity of the circular law for random matrices with non i.i.d. entries. Let A be a random n \times n real matrix having as a random vector in R^{n^2} a log-concave isotropic unconditional law. In particular, the entries are…

Exact evaluation of $<{\rm Tr} S^p>$ is here performed for real symmetric matrices $S$ of arbitrary order $n$, up to some integer $p$, where the matrix entries are independent identically distributed random variables, with an arbitrary…

A novel lower bound is introduced for the full rank probability of random finite field matrices, where a number of elements with known location are identically zero, and remaining elements are chosen independently of each other, uniformly…

We determine the distribution of the sandpile group (a.k.a. Jacobian) of the Erd\H{o}s-R\'enyi random graph G(n,q) as n goes to infinity. Since any particular group appears with asymptotic probability 0 (as we show), it is natural ask for…

We obtain lower tail estimates for the smallest singular value of random matrices with independent but non-identically distributed entries. Specifically, we consider $n\times n$ matrices with complex entries of the form \[ M = A\circ X + B…

We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This…

Let $R$ be a finite local ring. We prove a quantitative universality statement for the cokernel of random matrices with i.i.d. entries valued in $R$. Rather than use the moment method, we use the Lindeberg replacement technique. This…

In this paper, we investigate the eigenvalue distribution of a class of kernel random matrices whose $(i,j)$-th entry is $f(X_i,X_j)$ where $f$ is a symmetric function belonging to the Paley-Wiener space $\mathcal{B}_c$ and $(X_i)_{1\leq i…

We analyze the empirical spectral distribution of random periodic band matrices with correlated entries. The correlation structure we study was first introduced in 2015 by Hochst\"attler, Kirsch and Warzel, who named their setup "almost…

We study the rank of the random $n\times m$ 0/1 matrix ${\bf A}_{n,m;k}$ where each column is chosen independently from the set $\Omega_{n,k}$ of 0/1 vectors with exactly $k$ 1's. Here 0/1 are the elements of the field $GF_2$. We obtain an…

We consider the binomial random set model $[n]_p$ where each element in $\{1,\dots,n\}$ is chosen independently with probability $p:=p(n)$. We show that for essentially all regimes of $p$ and very general conditions for a matrix $A$ and a…

The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the…

Let $A$ be the adjacency matrix of a uniformly random $d$-regular digraph on $n$ vertices, and suppose that $\min(d,n-d)\geq\lambda n$. We show that for any $\kappa \geq 0$, \[\mathbb{P}[s_n(A)\leq\kappa]\leq…

The Cohen-Lenstra heuristic predicts the distribution of ideal class groups over number fields. Random matrix models provide a natural framework for explaining this heuristic, and recent results demonstrate the effectiveness of these tools.…

We show that for an $n\times n$ random matrix $A$ with independent uniformly anti-concentrated entries, such that $\mathbb{E} ||A||^2_{HS}\leq K n^2$, the smallest singular value $\sigma_n(A)$ of $A$ satisfies $$ P\left( \sigma_n(A)\leq…

In this paper we establish the limit of the empirical spectral distribution of quaternion sample covariance matrices. Suppose $\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}$ is a quaternion random matrix. For each $n$, the entries…

We consider the moment space $\mathcal{M}_n$ corresponding to $p \times p$ real or complex matrix measures defined on the interval $[0,1]$. The asymptotic properties of the first $k$ components of a uniformly distributed vector $(S_{1,n},…