Related papers: Distances between Random Orthogonal Matrices and I…

We present several refinements on the fluctuations of sequences of random vectors (with values in the Euclidean space $\mathbb{R}^d$) which converge after normalization to a multidimensional Gaussian distribution. More precisely we refine…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D$ and valency $k \ge 3$. In [Homotopy in $Q$-polynomial distance-regular graphs, Discrete Math., {\bf 223} (2000), 189-206], H. Lewis showed that the girth of…

The Gilbert graph $\text{Gilbert}(q,n,d)$, which arises naturally in graph theory and coding theory, is the regular graph on $\mathbb{F}_q^n$ in which two vertices are adjacent if their Hamming distance is less than $d$, and it is…

In a recent paper the author proved a theorem to the effect that the matrix of normalized Euclidean distances on the set of specially distributed random points in the $n$-dimensional Euclidean space $\mathbb R^{n}$ with independent…

Let $(P_N)_{N\ge0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\ldots, z_{N,N}$ of $P_N$ we derive lower estimates for $\min_{i\ne j}|z_{i,N}-z_{j,N}|$ and…

Consider an $N\times N$ hermitian random matrix with independent entries, not necessarily Gaussian, a so called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between…

This work considers the asymptotic behavior of the distance between two sample covariance matrices (SCM). A general result is provided for a class of functionals that can be expressed as sums of traces of functions that are separately…

A real symmetric matrix $A$ is said to be completely positive if it can be written as $BB^t$ for some (not necessarily square) nonnegative matrix $B$. A simple graph $G$ is called a completely positive graph if every doubly nonnegative…

We study the problem of recovering a globally consistent Euclidean embedding of data, given only a local distance graph and propose a method that optimally represents these distances. The method operates solely on a neighborhood graph…

For an $n \times n$ independent-entry random matrix $X_n$ with eigenvalues $\lambda_1, \ldots, \lambda_n$, the seminal work of Rider and Silverstein asserts that the fluctuations of the linear eigenvalue statistics $\sum_{i=1}^n…

Let $G_n=\mathbb{Z}_n\times \mathbb{Z}_n$ for $n\geq 4$ and $S=\{(i,0),(0,i),(i,i): 1\leq i \leq n-1\}\subset G_n$. Define $\Gamma(n)$ to be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is known that $\Gamma(n)$ is a…

Consider the Euclidean traveling salesman problem with $n$ random points on the plane. Suppose that one of the points is shifted to a new random location. This gives us a new optimal path. Consider such shifts for each of the $n$ points. Do…

In this paper we study higher-order difference equations which can be written as follows: $$ \mathbf{J} (y_0,y_1,...)^T = \lambda^N (y_0,y_1,...)^T, $$ where $\mathbf{J}$ is a $(2N+1)$-diagonal bounded banded matrix…

We establish quantitative comparisons between classical distances for probability distributions belonging to the class of convex probability measures. Distances include total variation distance, Wasserstein distance, Kullback-Leibler…

We prove that a very general class of $N\times N$ Hermitian random band matrices is in the delocalized phase when the band width $W$ exceeds the critical threshold, $W\gg \sqrt{N}$. In this regime, we show that, in the bulk spectrum, the…

Given any two vertices u, v of a random geometric graph, denote by d_E(u,v) their Euclidean distance and by d_G(u,v) their graph distance. The problem of finding upper bounds on d_G(u,v) in terms of d_E(u,v) has received a lot of attention…

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_\text{c}(M)$ of compactly supported diffeomorphisms, for various Sobolev norms $W^{s,p}$. Our main result is that the geodesic distance…

Consider an $n \times n$ matrix polynomial $P(\lambda)$. A spectral norm distance from $P(\lambda)$ to the set of $n \times n$ matrix polynomials that have a given scalar $\mu\in\mathbb{C}$ as a multiple eigenvalue was introduced and…

We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups $G$ and $H$. We prove some general structural results on how the distance behaves with respect to natural group…

We consider a primitive distance-regular graph $\Gamma$ with diameter at least $3$. We use the intersection numbers of $\Gamma$ to find a positive semidefinite matrix $G$ with integer entries. We show that $G$ has determinant zero if and…