Related papers: Condition Numbers of Gaussian Random Matrices

A matrix is called totally positive if every minor of it is positive. Such matrices are well studied and have numerous applications in Mathematics and Computer Science. We study how many times the value of a minor can repeat in a totally…

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists and is a constant strictly…

A necessary condition is given for a sequence of identically distributed and pairwise positively quadrant dependent random variables obeying the strong laws of large numbers with respect to the normalising constants $n^{1/p}$ $(1 \leqslant…

This paper investigates the nonasymptotic properties of the spectral norm of some random matrices with independent columns. In particular, we consider an $m\times n$ random matrix $BA$, where $A$ is an $N\times n$ random matrix with…

Let $\Omega\subset\mathbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega=-\Omega$ with fixed Gaussian volume. Let $A=A_{x}$ be the second fundamental form of $\partial\Omega$ at $x$, i.e. $A$ is the matrix of…

Studying conditional independence among many variables with few observations is a challenging task. Gaussian Graphical Models (GGMs) tackle this problem by encouraging sparsity in the precision matrix through $l_q$ regularization with…

We analyze the probability that a random m-dimensional linear subspace of R^n both intersects a regular closed convex cone C\subseteq R^n and lies within distance \alpha of an m-dimensional subspace not intersecting C (except at the…

We consider the detection problem of correlations in a $p$-dimensional Gaussian vector, when we observe $n$ independent, identically distributed random vectors, for $n$ and $p$ large. We assume that the covariance matrix varies in some…

We study the RPA equations in their most general form by taking the matrix elements appearing in the RPA equations as random. This yields either a unitarily or an orthogonally invariant random-matrix model which is not of the Cartan type.…

Motivated by the general matrix deviation inequality for i.i.d ensemble Gaussian matrix, we study its universality property. As a starting point for this problem, we show that this property holds for $\ell_{p}$-norm with $1\leq p< \infty$…

We consider the infinite sequences $(A\_n)\_{n\in\NN}$ of $2\times2$ matrices with nonnegative entries, where the $A\_n$ are taken in a finite set of matrices. Given a vector $V=\pmatrix{v\_1\cr v\_2}$ with $v\_1,v\_2>0$, we give a…

A central question in random matrix theory is universality. When an emergent phenomena is observed from a large collection of chosen random variables it is natural to ask if this behavior is specific to the chosen random variable or if the…

We study propagation of the RegularGcc global constraint. This ensures that each row of a matrix of decision variables satisfies a Regular constraint, and each column satisfies a Gcc constraint. On the negative side, we prove that…

Products of $M$ i.i.d. random matrices of size $N \times N$ are related to classical limit theorems in probability theory ($N=1$ and large $M$), to Lyapunov exponents in dynamical systems (finite $N$ and large $M$), and to universality in…

Using a Coulomb gas technique, we compute analytically the probability $\mathcal{P}_\beta^{(C)}(N_+,N)$ that a large $N\times N$ Cauchy random matrix has $N_+$ positive eigenvalues, where $N_+$ is called the index of the ensemble. We show…

Given a constraint satisfaction problem (CSP) on $n$ variables, $x_1, x_2, \dots, x_n \in \{\pm 1\}$, and $m$ constraints, a global cardinality constraint has the form of $\sum_{i = 1}^{n} x_i = (1-2p)n$, where $p \in (\Omega(1), 1 -…

We prove that there is a universal constant $C>0$ with the following property. Suppose that $n\in \mathbb{N}$ and that $\mathsf{A}=(a_{ij})\in M_n(\mathbb{R})$ is a symmetric stochastic matrix. Denote the second-largest eigenvalue of…

A matrix is homogeneous if all of its entries are equal. Let $P$ be a $2\times 2$ zero-one matrix that is not homogeneous. We prove that if an $n\times n$ zero-one matrix $A$ does not contain $P$ as a submatrix, then $A$ has an $cn\times…

We consider a class of sparse random matrices of the form $A_n =(\xi_{i,j}\delta_{i,j})_{i,j=1}^n$, where $\{\xi_{i,j}\}$ are i.i.d.~centered random variables, and $\{\delta_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$…

Let $\textrm{Mat}_2(\mathbb{R})$ be the set of $2 \times 2$ matrices with real entries. For any $\varepsilon>0$ and any finitely--supported probability measure $\mu$ on $\textrm{Mat}_2(\mathbb{R})$, we prove that either \[ T(\mu) = \sum_{X,…