Related papers: Rank fluctuations of matrix products and a moment …
For random integer matrices $M_1,\ldots,M_k \in \operatorname{Mat}_n(\mathbb{Z})$ with independent entries, we study the distribution of the cokernel $\operatorname{cok}(M_1 \cdots M_k)$ of their product. We show that this distribution…
We consider uniformly random strictly upper-triangular matrices in $\operatorname{Mat}_n(\mathbb{F}_q)$. For such a matrix $A_n$, we show that $n-\operatorname{rank}(A_n) \approx \log_q n$ as $n \to \infty$, and find that the fluctuations…
In this paper, we study the combinatorial relations between the cokernels $\text{cok}(A_n+px_iI_n)$ ($1 \le i \le m$) where $A_n$ is an $n \times n$ matrix over the ring of $p$-adic integers $\mathbb{Z}_p$, $I_n$ is the $n \times n$…
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…
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…
Given a prime $p$, let $P(t)$ be a non-constant monic polynomial in $t$ over the ring $\mathbb{Z}_{p}$ of $p$-adic integers. Let $X_{n}$ be an $n \times n$ random matrix over $\mathbb{Z}_{p}$ with independent entries that lie in any residue…
In this paper, we study the distribution of the cokernels of random $p$-adic matrices with fixed zero entries. Let $X_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ in which some entries are fixed to be zero and the other entries…
In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit…
We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index $\beta = 1,2,4$ respectively) where time corresponds to the number of terms in the…
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 $k_n$,…
In this article, the joint fluctuations of the extreme eigenvalues and eigenvectors of a large dimensional sample covariance matrix are analyzed when the associated population covariance matrix is a finite-rank perturbation of the identity…
We study the random sequential adsorption of $k$-mers on the fully-connected lattice with $N=kn$ sites. The probability distribution $T_n(s,t)$ of the time $t$ needed to cover the lattice with $s$ $k$-mers is obtained using a generating…
We use the moment method of Wood to study the distribution of random finite modules over a countable Dedekind domain with finite quotients, generated by taking cokernels of random $n\times n$ matrices with entries valued in the domain.…
Covariances and variances of linear statistics of a point process can be written as integrals over the truncated two-point correlation function. When the point process consists of the eigenvalues of a random matrix ensemble, there are often…
Distributed consensus and other linear systems with system stochastic matrices $W_k$ emerge in various settings, like opinion formation in social networks, rendezvous of robots, and distributed inference in sensor networks. The matrices…
We study the scattering of waves in systems with losses or gains simulated by imaginary potentials. This is done for a complex delta potential that corresponds to a spatially localized absorption or amplification. In the Argand plane the…
We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models,…
A recent approach to the Beck-Fiala conjecture, a fundamental problem in combinatorics, has been to understand when random integer matrices have constant discrepancy. We give a complete answer to this question for two natural models:…
We investigate uniform random block lower bidiagonal matrices over the finite field $\mathbb{F}_q$, and prove that their rank undergoes a phase transition. First, we consider block lower bidiagonal matrices with $(k_n+1)\times k_n$ blocks…
We investigate moment sequences of probability measures on $E\subset\mathbb{R}$ under constraints of certain moments being fixed. This corresponds to studying sections of $n$-th moment spaces, i.e. the spaces of moment sequences of order…