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Singularity of random Bernoulli matrices
For each $n$, let $M_n$ be an $n\times n$ random matrix with independent $\pm 1$ entries. We show that ${\mathbb P}\{\mbox{$M_n$ is singular}\}=(1/2+o_n(1))^n$, which settles an old problem. Some
The smallest singular value of inhomogeneous square random matrices
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 $A$
Sample covariance matrices of heavy-tailed distributions
Let $p>2$, $B\geq 1$, $N\geq n$ and let $X$ be a centered $n$-dimensional random vector with the identity covariance matrix such that $\sup\limits_{a\in S^{n-1}}{\mathrm E}|\langle X,a\rangle|^p\leq
Outliers in spectrum of sparse Wigner matrices
A non-centered counterpart of the theorem allows to obtain asymptotic expressions for eigenvalues of the Erdős--Renyi graphs, which were unknown in the regime $n p_n=\Theta(\log n)$.
Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries
Let A = (aij) be an n × n random matrix with i.i.d. entries such that Ea11 = 0 and Ea112 = 1. We prove that for any δ > 0 there is L > 0 depending only on δ, and a subset N of B2n of cardinality at
Singularity of sparse Bernoulli matrices
Let $M_n$ be an $n\times n$ random matrix with i.i.d. Bernoulli(p) entries. We show that there is a universal constant $C\geq 1$ such that, whenever $p$ and $n$ satisfy $C\log n/n\leq p\leq C^{-1}$,
A probabilistic version of Rosenthal’s inequality
k=1 m{t ∈ [0, 1] : |fk(t)| > τ} (τ > 0), where m is the Lebesgue measure. Let F ∗(t) be the non-increasing left-continuous rearrangement of F (t) and, as usual, χA be the characteristic function of a
The smallest singular value of random rectangular matrices with no moment assumptions on entries
Let δ > 1 and β > 0 be some real numbers. We prove that there are positive u, v, N0 depending only on β and δ with the following property: for any N,n such that N ≥ max(N0, δn), any N × n random