Related papers: On the Largest Singular Value/Eigenvalue of a Rand…

Let $A$ be an $n\times n$ real matrix, and let $M$ be an $n\times n$ random matrix whose entries are i.i.d sub-Gaussian random variables with mean $0$ and variance $1$. We make two contributions to the study of $s_n(A+M)$, the smallest…

This paper discusses the computation of real Z-eigenvalues and H-eigenvalues of nonsymmetric tensors. A general nonsymmetric tensor has finitely many Z-eigenvalues, while there may be infinitely many ones for special tensors. In the…

We study the small ball probability of an order-$\ell$ simple random tensor $X=X^{(1)}\otimes\cdots\otimes X^{(\ell)}$ where $X^{(i)}, 1\leq i\leq\ell$ are independent random vectors in $\mathbb{R}^n$ that are log-concave or have…

Consider a sum $S_n=v_i\varepsilon_1+\cdots+v_n\varepsilon_{n}$, where $(v_i)^{n}_{i=1}$ are non-zero vectors in $\mathbb{R}^{d}$ and $(\varepsilon_i)^{n}_{i=1}$ are independent Rademacher random variables (i.e.,…

In this article, we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let $\mathcal{G}$ denote a random tensor of order $r$ and dimension $n$ drawn from the density \[ f(\mathcal{G}) \propto…

We take a first small step to extend the validity of Rudelson-Vershynin type estimates to some sparse random matrices, here random permutation matrices. We give lower (and upper) bounds on the smallest singular value of a large random…

Stationary and isotropic iteration stable random tessellations are considered, which can be constructed by a random process of cell division. The collection of maximal polytopes at a fixed time $t$ within a convex window $W\subset{\Bbb…

We obtain a non-asymptotic bound for the expected injective norm of a random tensor with independent entries. This bound is similar to the bound by Bandeira and van Handel (2016) for the expected spectral norm of a random matrix with…

The maximum of the absolute value of a real homogeneous polynomial of degree $d\ge 3$ on the unit sphere corresponds to the spectral norm of the induced real $d$-symmetric tensor $\mathcal{S}$. We give two sequences of upper bounds on the…

We study the component structure of the random graph $G=G_{n,m,d}$. Here $d=O(1)$ and $G$ is sampled uniformly from ${\mathcal G}_{n,m,d}$, the set of graphs with vertex set $[n]$, $m$ edges and maximum degree at most $d$. If $m=\mu n/2$…

For random $d$-regular graphs on $N$ vertices with $1 \ll d \ll N^{2/3}$, we develop a $d^{-1/2}$ expansion of the local eigenvalue distribution about the Kesten-McKay law up to order $d^{-3}$. This result is valid up to the edge of the…

Let $M_n$ denote a random symmetric $n \times n$ matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely…

We consider the spectral gap of a uniformly chosen random $(d_1,d_2)$-biregular bipartite graph $G$ with $|V_1|=n, |V_2|=m$, where $d_1,d_2$ could possibly grow with $n$ and $m$. Let $A$ be the adjacency matrix of $G$. Under the assumption…

Let $R_n$ be a $n \times n$ random matrix with i.i.d. subgaussian entries. Let $M$ be a $n \times n$ deterministic matrix with norm $\lVert M \rVert \le n^\gamma$ where $1/2<\gamma<1$. The goal of this paper is to give a general estimate of…

We study the probability that all eigenvalues of the Laguerre unitary ensemble of n by n matrices are between 0 and t, i.e., the largest eigenvalue distribution. Associated with this probability, in the ladder operator approach for…

We obtain an optimal bound for a Gaussian approximation of a large class of vector-valued random processes. Our results provide a substantial generalization of earlier results that assume independence and/or stationarity. Based on the decay…

For a pair of coupled rectangular random matrices we consider the squared singular values of their product, which form a determinantal point process. We show that the limiting mean distribution of these squared singular values is described…

We prove upper bounds on the number of resonances and eigenvalues of Schr\"odinger operators $-\Delta+V$ with complex-valued potentials, where $d\geq 3$ is odd. The novel feature of our upper bounds is that they are \emph{effective}, in the…

Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine the number of eigenvectors and eigenvalues of a generic tensor, and we show that the…

We give simple criteria to identify the exponential order of magnitude of the absolute value of the determinant for wide classes of random matrix models, not requiring the assumption of invariance. These include Gaussian matrices with…