Related papers: On sparse random combinatorial matrices

Here we introduce the non-Hermitian diluted banded random matrix (nHdBRM) ensemble as the set of $N\times N$ real non-symmetric matrices whose entries are independent Gaussian random variables with zero mean and variance one if $|i-j|<b$…

We show that a random subset of the rank-$n$ projective geometry $\text{PG}(n-1,q)$ is, with high probability, not $(b,c)$-decomposable: if $k$ is its colouring number, it does not admit a partition of its ground set into classes of size at…

Let $M$ be a random $m \times n$ matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly at random given its weight. Let $N(n,m)$ denote…

Let $f:\{-1,1\}^n$ be a polynomial with at most $s$ non-zero real coefficients. We give an algorithm for exactly reconstructing f given random examples from the uniform distribution on $\{-1,1\}^n$ that runs in time polynomial in $n$ and…

A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graph $G_{n,1/2}$ typically admits a covering of a constant fraction of its edges by edge-disjoint, nearly maximum cliques. We show that this is not the case. The…

Let $M$ be a random matrix chosen according to Haar measure from the unitary group $\mathrm{U}(n,\mathbb{C})$. Diaconis and Shahshahani proved that the traces of $M,M^2,\ldots,M^k$ converge in distribution to independent normal variables as…

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erd\H{o}s-R\'enyi graphs. Our error bounds are sharp for a large class of sparse random matrices.…

We consider the set $\mathcal M_n(\mathbb Z; H)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain upper bounds on the number of matrices from $\mathcal M_n(\mathbb Z; H)$, for which the characteristic polynomial…

Let $M_n$ be a random matrix of size $n\times n$ and let $\lambda_1,...,\lambda_n$ be the eigenvalues of $M_n$. The empirical spectral distribution $\mu_{M_n}$ of $M_n$ is defined as $$\mu_{M_n}(s,t)=\frac{1}{n}# \{k\le n, \Re(\lambda_k)\le…

We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a $2\times 2$ complex matrix). We define the ensemble by the matrix probability distribution function that is…

Given a matrix $A$ with integer entries, a subset $S$ of an abelian group and $r \in \mathbb N$, we say that $S$ is $(A,r)$-Rado if any $r$-colouring of $S$ yields a monochromatic solution to the system of equations $Ax=0$. A classical…

We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all…

Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $n\times n$ array $A$…

A well-known application of the dependent random choice asserts that any $n$-vertex graph $G$ with positive edge density contains a `rich' vertex subset $U$ of size $n^{1-o(1)}$ such that every pair of vertices in $U$ has at least…

We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by…

Let $P_n^1,\dots, P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum_{\ell=1}^d P_n^\ell$. We show that if $\log^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral…

Symmetric matrices with zero row sums occur in many theoretical settings and in real-life applications. When the offdiagonal elements of such matrices are i.i.d. random variables and the matrices are large, the eigenvalue distributions…

We prove a quantitative deterministic equivalence theorem for the logarithmic potentials of deterministic complex $N\times N$ matrices subject to small random perturbations. We show that with probability close to $1$ this log-potential is,…

We show that for an $n\times n$ random matrix $A$ with independent uniformly anti-concentrated entries, such that $\mathbb{E} ||A||^2_{HS}\leq K n^2$, the smallest singular value $\sigma_n(A)$ of $A$ satisfies $$ P\left( \sigma_n(A)\leq…

The Thue--Morse sequence $\{t(n)\}_{n\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp.…