Related papers: Random integral matrices and the Cohen Lenstra Heu…

In this paper we study the cokernels of various random integral matrix models, including random symmetric, random skew-symmetric, and random Laplacian matrices. We provide a systematic method to establish universality under very general…

We formulate a model for the average behaviour of ray class groups of real quadratic fields with respect to a fixed rational modulus, locally at a finite set $S$ of odd primes. To that end, we introduce Arakelov ray class groups of a number…

In this note, we study the distribution of the rational canonical form of a random matrix over the finite field $\mathbb{F}_p$, whose entries are independent and $\epsilon$-balanced with $\epsilon\in(0,1-1/p]$. We show that, as the matrix…

Let $\mathbf{B}_n$ be an $n\times n$ Haar-uniform band matrix over $\mathbb{Z}_p$ with band width $w_n$. We prove that $\text{cok}(\mathbf{B}_n)$ has Cohen-Lenstra limiting distribution if and only if \[\lim_{n\to\infty}…

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 show that under reasonable conditions, a random $n\times (2+\epsilon) n$ integer matrix is surjective on $\mathbb{Z}^{n}$ with probability $1-O(e^{-cn})$. We also conjecture that this should hold for $n\times (1+\epsilon)n$, and provide…

We give an elementary proof of the statement that if an idempotent complete preadditive category has weak kernels and weak cokernels, then it has $n$-kernels if and only if it has $n$-cokernels, where $n$ is a nonnegative integer. As a…

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 extend the Cohen-Lenstra heuristics to the setting of ray class groups of imaginary quadratic number fields, viewed as exact sequences of Galois modules. By asymptotically estimating the mixed moments governing the distribution of a…

In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers $\mathcal{O}$ of a quadratic extension $K$ of $\mathbb{Q}_p$. For each positive integer $n$, let $X_n$ be a random $n…

Let $S=XX^T$ be the (unscaled) sample covariance matrix where $X$ is a real $p \times n$ matrix with independent entries. It is well known that if the entries of $X$ are independent and identically distributed (i.i.d.) with enough moments…

We prove that $\frac{\log n}{n}$ is the sharp threshold for universality of the distribution of cokernels of random matrices over $\mathbb{Z}_p$. More precisely, let $\alpha_n = \frac{c\log n}{n}$ for a constant $c>0$ and let $A(n)$ be an…

We prove universality for cokernels of random integral matrices with symmetries via an approach different from the classical surjection moment method introduced by Wood (arXiv:1402.5149). In the symmetric case, we reprove Hodges'…

We determine the distribution of the sandpile group (a.k.a. Jacobian) of the Erd\H{o}s-R\'enyi random graph G(n,q) as n goes to infinity. Since any particular group appears with asymptotic probability 0 (as we show), it is natural ask for…

Let g be a random element of a finite classical group G, and let \lambda_{z-1}(g) denote the partition corresponding to the polynomial z-1 in the rational canonical form of g. As the rank of G tends to infinity, \lambda_{z-1}(g) tends to a…

The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular distribution, has attracted much attention. The $2k$th moment of the limit equals…

We consider a class of random banded Hessenberg matrices with independent entries having identical distributions along diagonals. The distributions may be different for entries belonging to different diagonals. For a sequence of $n\times n$…

Let A be an n*n random matrix with mean zero and independent inhomogeneous non-constant subgaussian entries. We get that for any k<c\sqrt{n}, the probability of the matrix has a lower rank than n-k that is sub-exponential. Furthermore, we…

In [NVP22], Nguyen and Van Peski raised the question of whether the surjective flag of $\mathbb Z_p$-modules modeled by $\mathrm{cok}(M_1\cdots M_k)\twoheadrightarrow \dots\twoheadrightarrow \mathrm{cok}(M_1)$ for independent random…

Let $L/K$ be a quadratic extension of global fields. We study Cohen-Lenstra heuristics for the $\ell$-part of the relative class group $G_{L/K} := \textrm{Cl}(L/K)$ when $K$ contains $\ell^n$th roots of unity. While the moments of a…