Related papers: Random integral matrices: universality of surjecti…

For $n \times n$ random integer matrices $M_1,\ldots,M_k$, the cokernels of the partial products $\mathrm{cok}(M_1 \cdots M_i), 1 \leq i \leq k$ naturally define a random flag of abelian $p$-groups. We prove that as $n \to \infty$, this…

We show that, under mild assumptions, the spectrum of a sum of independent random matrices is close to that of the Gaussian random matrix whose entries have the same mean and covariance. This nonasymptotic universality principle yields…

We consider a space of sparse Boolean matrices of size $n \times n$, which have finite co-rank over $GF(2)$ with high probability. In particular, the probability such a matrix has full rank, and is thus invertible, is a positive constant…

Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in…

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime $p$ and a positive integer $n$, consider a random $n \times n$ matrix $X_n$ over the ring…

Let X_R be the zero locus in RP^n of one or two independently and Weyl distributed random real quadratic forms (this is the same as requiring that the corresponding symmetric matrices are in the Gaussian Orthogonal Ensemble). We prove that…

We consider certain large random matrices, called random inner-product kernel matrices, which are essentially given by a nonlinear function $f$ applied entrywise to a sample-covariance matrix, $f(X^TX)$, where $X \in \mathbb{R}^{d \times…

In this paper, we study the joint distribution of the cokernels of random $p$-adic matrices. Let $p$ be a prime and $P_1(t), \cdots, P_l(t) \in \mathbb{Z}_p[t]$ be monic polynomials whose reductions modulo $p$ in $\mathbb{F}_p[t]$ are…

We investigate the universality of microscopic eigenvalue correlations for Random Matrix Theories with the global symmetries of the QCD partition function. In this article we analyze the case of real valued chiral Random Matrix Theories…

We study the distribution of the cokernels of adjacency matrices (the Smith groups) of certain models of random $r$-regular graphs and directed graphs, using recent mixing results of M\'esz\'aros. We explain how convergence of such…

We study the distribution of the cokernels of random row-sparse integral matrices $A_n$ according to the determinantal measure from a structured matrix $B_n$ with a parameter $k_n \ge 3$. Under a mild assumption on the growth rate of $k_n$,…

The probability that a tuple of matrices together with all scalars generates a finite incidence ring is calculated. It is proved that all real and complex finite-dimensional incidence algebras are generated by two randomly chosen matrices.

Let $A$ be drawn uniformly at random from the set of all $n\times n$ symmetric matrices with entries in $\{-1,1\}$. We show that \[ \mathbb{P}( \det(A) = 0 ) \leq e^{-cn},\] where $c>0$ is an absolute constant, thereby resolving a…

We consider injectivity and surjectivity of some maps on the exterior algebra of isomorphic finite-dimensional vector spaces. We prove the properties of the maps in full generality, for any dimension of the vector space and any subspace. We…

We use the steepest descents method to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution P_{N}(z_{1},...,z_{N}) = Z_{N}^{-1} e^{-N\Sigma_{i=1}^{N}V_{\alpha}(z_{i})}…

It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this…

We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance $N^{-3/4+\beta}$ for some positive…

We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving…

In this paper the kernel for the spectral correlation functions of the invariant chiral random matrix ensembles with real ($\beta =1$) and quaternion real ($\beta = 4$) matrix elements is expressed in terms of the kernel of the…

Let $(R, \mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/\mathfrak{m} = \mathbb{F}_{q}$. Given a monic polynomial $P(t) \in R[t]$ whose reduction modulo $\mathfrak{m}$ gives an irreducible polynomial…