Related papers: The circular law for sparse random combinatorial m…

We consider the random matrix obtained by picking vectors randomly from a large collection of mutually unbiased bases of $\mathbb{C}^n$, and prove that the spectral distribution converges to the Marchenko-Pastur law. This shows that vectors…

Let $x_i$, $i\in\mathbb{Z}$ be a sequence of i.i.d. standard normal random variables. Consider rectangular Toeplitz $\mathbf{X}=\left(x_{j-i}\right)_{1\leq i\leq p,1\leq j\leq n}$ and circulant $\mathbf{X}=\left(x_{(j-i)\mod…

We prove an estimate on the smallest singular value of a multiplicatively and additively deformed random rectangular matrix. Suppose $n\le N \le M \le \Lambda N$ for some constant $\Lambda \ge 1$. Let $X$ be an $M\times n$ random matrix…

In this paper, we study the restricted isometry property of partial random circulant matrices. For a bounded subgaussian generator with independent entries, we prove that the partial random circulant matrices satisfy $s$-order RIP with high…

We study the volume of rigid loop-$O(n)$ quadrangulations with a boundary of length $2p$ in the non-generic critical regime. We prove that, as the half-perimeter $p$ goes to infinity, the volume scales in distribution to an explicit random…

We develop a theoretical approach to compute the conditioned spectral density of $N \times N$ non-invariant random matrices in the limit $N \rightarrow \infty$. This large deviation observable, defined as the eigenvalue distribution…

We study cycle counts in permutations of $1,\dots,n$ drawn at random according to the Mallows distribution. Under this distribution, each permutation $\pi \in S_n$ is selected with probability proportional to $q^{\text{inv}(\pi)}$, where…

Let $\{x_{\alpha}\}_{\alpha \in \mathbb{Z}}$ and $\{y_{\alpha}\}_{\alpha \in \mathbb{Z}}$ be two independent collections of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a nonsymmetric…

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…

Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/p_n$ is bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i<j\leq…

The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate…

We establish that the limiting spectral distribution of a block-rescaled empirical covariance matrix is an arcsine law when the ratio between the dimension and the underlying sample size converges to 1 and when the samples corresponding to…

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For…

In this paper we consider a new normalization of matrices obtained by choosing distinct codewords at random from linear codes over finite fields and find that under some natural algebraic conditions of the codes their empirical spectral…

We consider the empirical eigenvalue distribution for a class of non-Hermitian random block tridiagonal matrices $T$ with independent entries. The matrix has $n$ blocks on the diagonal and each block has size $\ell_n$, so the whole matrix…

We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is…

In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main…

We revisit the asymptotic analysis of probabilistic construction of adjacency matrices of expander graphs proposed in [4]. With better bounds we derived a new reduced sample complexity for the number of nonzeros per column of these…

Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = \delta_{ij} \xi_{ij}$ for $i \leq j$. $\delta_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq…

Let $M_n$ be drawn uniformly from all $\pm 1$ symmetric $n \times n$ matrices. We show that the probability that $M_n$ is singular is at most $\exp(-c(n\log n)^{1/2})$, which represents a natural barrier in recent approaches to this…