Distribution of the cokernels of determinantal row-sparse matrices

TL;DR

This paper investigates the distribution of cokernels of random row-sparse matrices and proves their convergence to Cohen--Lenstra distributions under certain growth conditions, extending previous results to all primes.

Contribution

It generalizes earlier work by establishing convergence to Cohen--Lenstra distributions for a broader class of matrices and primes, under mild growth assumptions.

Findings

Distribution of cokernels converges to Cohen--Lenstra for all primes

Extends previous results from specific primes to all primes

Applicable to matrices with growing parameter k_n

Abstract

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 $k_n$, we prove that the distribution of the $p$-Sylow subgroup of the cokernel of $A_n$ converges to that of Cohen--Lenstra for every prime $p$. Our result extends the work of A. M\'esz\'aros which established convergence to the Cohen--Lenstra distribution when $p \ge 5$ and $k_n=3$ for all positive integers $n$.