Related papers: The smallest singular value of random rectangular …

This work shows that the smallest natural number $d_n$ that is not the determinant of some $n\times n$ binary matrix is at least $c\,2^n/n$ for $c=1/201$. That same quantity naturally lower bounds the number of distinct integers $D_n$ which…

Let $X$ be a symmetric, isotropic random vector in $\mathbb{R}^m$ and let $X_1...,X_n$ be independent copies of $X$. We show that under mild assumptions on $\|X\|_2$ (a suitable thin-shell bound) and on the tail-decay of the marginals…

The statistics of the smallest eigenvalue of Wishart-Laguerre ensemble is important from several perspectives. The smallest eigenvalue density is typically expressible in terms of determinants or Pfaffians. These results are of utmost…

We are concerned with the small ball behavior of the smallest singular value of random matrices. Often, establishing such results involves, in some capacity, a discretization of the unit sphere. This requires bounds on the norm of the…

The sum-product conjecture of Erd\H os and Szemer\'edi states that, given a finite set $A$ of positive numbers, one can find asymptotic lower bounds for $\max\{|A+A|,|A\cdot A|\}$ of the order of $|A|^{1+\delta}$ for every $\delta <1$. In…

In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the $1/N$ expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives…

For a $d$-dimensional random vector $X$, let $p_{n, X}(\theta)$ be the probability that the convex hull of $n$ independent copies of $X$ contains a given point $\theta$. We provide several sharp inequalities regarding $p_{n, X}(\theta)$ and…

Given a graph $G = (V,E)$, an $(\alpha, \beta)$-ruling set is a subset $S \subseteq V$ such that the distance between any two vertices in $S$ is at least $\alpha$, and the distance between any vertex in $V$ and the closest vertex in $S$ is…

We compute exact asymptotic results for the probability of the occurrence of large deviations of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we…

For a positive rational $\alpha$, call a set of distinct positive integers $\{a_1, a_2, \ldots, a_r\}$ an $\alpha$-partition of $n$, if the sum of the $a_i$ is equal to $n$ and the sum of the reciprocals of the $a_i$ is equal to $\alpha$.…

This paper studies the problem of selecting a submatrix of a positive definite matrix in order to achieve a desired bound on the smallest eigenvalue of the submatrix. Maximizing this smallest eigenvalue has applications to selecting input…

We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $\beta$-ensembles in three cases : the case of Wigner matrices without Gaussian tails, that is Wigner…

Let $\Delta(n)$ denote the smallest positive integer $m$ such that $a^3+a(1\le a\le n)$ are pairwise distinct modulo $m$. The purpose of this paper is to determine $\Delta(n)$ for all positive integers $n$.

Let $U$ be a Morse function on a compact connected $m$-dimensional Riemannian manifold, $m \geq 2,$ satisfying $\min U=0$ and let $\mathcal{U} = \{x \in M \: : U(x) = 0\}$ be the set of global minimizers. Consider the stochastic algorithm…

Let A be an n x n symmetric random matrix whose upper-triangular entries are independent and follow possibly non-identical subgaussian distributions. This paper investigates the spectral properties of A, including its eigenvalues and…

Let G=SL_n. Let K=Z/pZ, p a prime. Let A\subset G(K) generate G(K). Suppose that |A|<p^{n+1-\delta}, delta>0. Then |A A A|>>|A|^{1+\epsilon}, where epsilon>0 and the implied constant depend only on n and delta.

For $g < n$, let $b\_1,...,b\_{n-g}$ be $n - g$ independent vectors in $\mathbb{R}^n$ with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is…

Let $X^\beta$ be a real symmetric or complex Hermitian matrix whose entries are independent Gaussian random fields. We provide the sufficient and necessary conditions such that multiple collisions of eigenvalue processes of $A^\beta +…

The bad science matrix problem consists in finding, among all matrices $A \in \mathbb{R}^{n \times n}$ with rows having unit $\ell^2$ norm, one that maximizes $\beta(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main…

One important question in the theory of lattices is to detect a shortest vector: given a norm and a lattice, what is the smallest norm attained by a non-zero vector contained in the lattice? We focus on the infinity norm and work with…