Related papers: On the permanent of a random symmetric matrix

We present a simple proof to a fact recently established in [5]: let $\xi$ be a symmetric random variable that has variance $1$, let $\Gamma=(\xi_{ij})$ be an $N \times n$ random matrix whose entries are independent copies of $\xi$, and set…

In this paper we report on new results relating to a conjecture regarding properties of $n\times n$, $n\leq 6$, positive definite matrices. The conjecture has been proven for $n\leq 4$ using computer-assisted sum of squares (SoS) methods…

We exhibit a randomized algorithm which given a matrix $A\in \mathbb{C}^{n\times n}$ with $\|A\|\le 1$ and $\delta>0$, computes with high probability an invertible $V$ and diagonal $D$ such that $\|A-VDV^{-1}\|\le \delta$ using…

Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an $n\times n$ matrix $M$ and will receive $n$ column-vectors of size $n$, denoted by $v_1,\ldots,v_n$, one by one. After seeing each vector $v_i$, we…

One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix…

Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq 3$, and denote the eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We prove that the…

Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*} {\mathbb P}(\text{$A$ is singular in…

Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\phi$ be the characteristic polynomial. Conditionally on the extended…

Let $\mathscr{U}(n,\tau)$ be the set of all {\rm(0,1)}-matrices of order $n$ with exactly $\tau$ 0's. Brualdi et al. investigated the maximum permanents of all matrices in $\mathscr{U}(n,\tau)$(R.A. Brualdi, J.L. Goldwasser, T.S. Michael,…

We study two types of unique continuation properties for the higher order Schr\"{o}dinger equation with potential $$ i\partial_tu=(-\Delta_x)^mu+V(t,x)u,\quad(t,x)\in\mathbb{R}^{1+n},\,2\leq m\in\mathbb{N}_+. $$ The first one says if $u$…

The permanent of unitary matrices and their blocks has attracted increasing attention in quantum physics and quantum computation because of connections with the Hong-Ou-Mandel effect and the Boson Sampling problem. In that context, it would…

Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This paper…

In an earlier paper, we discussed the probability that the determinant of a matrix undergoes the least change upon perturbation of one of its elements, provided that most or all of the elements of the matrix are chosen at random and that…

We consider the equation R(n)=Q(n)+M(n) R(n-1), with random non-i.i.d. coefficients (Q(n),M(n)), and show that the distribution tails of the stationary solution to this equation are regularly varying at infinity.

We form a sequence of oblong matrices by evaluating an integrable vector-valued function along the orbit of an ergodic dynamical system. We obtain an almost sure asymptotic result for the permanents of those matrices. We also give an…

Celebrated work of Jerrum, Sinclair, and Vigoda has established that the permanent of a {0,1} matrix can be approximated in randomized polynomial time by using a rapidly mixing Markov chain. A separate strand of the literature has pursued…

The rank of an n x n matrix A is equal to the size of its largest square submatrix with a nonzero determinant, and it can be computed in O(n^2.37) time. Analogously, the size of the largest square submatrix with nonzero permanent is defined…

Let $A$ be an $n\times n$ random symmetric matrix with independent identically distributed subgaussian entries of unit variance. We prove the following large deviation inequality for the rank of $A$: for all $1\leq k\leq c\sqrt{n}$,…

Let $A$ be an $n\times n$ random matrix with independent, identically distributed mean 0, variance 1 subgaussian entries. We prove that $$ \mathbb{P}(A\text{ has distinct singular values})\geq 1-e^{-cn} $$ for some $c>0$, confirming a…

Let $A=(a_{ij})$ be an $n$-by-$n$ matrix. For any real number $\mu$, we define the polynomial $$P_\mu(A)=\sum_{\sigma\in S_n} a_{1\sigma(1)}\cdots a_{n\sigma(n)}\,\mu^{\ell(\sigma)}\; ,$$ as the $\mu$-permanent of $A$, where $\ell(\sigma)$…