Cohen-Lenstra distribution for sparse matrices with determinantal biasing

TL;DR

This paper proves that the distribution of the p-Sylow subgroup of the cokernel of a certain random matrix, biased by the squared determinant, follows Cohen-Lenstra heuristics, linking to conjectures about random hypertrees.

Contribution

It establishes the Cohen-Lenstra distribution for the cokernel of a specific class of determinantal-biased sparse matrices.

Findings

Distribution of the p-Sylow subgroup matches Cohen-Lenstra heuristics.

Asymptotic behavior of the cokernel is characterized.

Supports conjecture relating random hypertrees to Cohen-Lenstra distribution.

Abstract

Let us consider the following matrix $B_n$. The columns of $B_n$ are indexed with $[n]=\{1,2,\dots,n\}$ and the rows are indexed with $[n]^3$. The row corresponding to $(x_1,x_2,x_3)\in [n]^3$ is given by $\sum_{i=1}^3 e_{x_i}$, where $e_1,e_2,\dots,e_n$ is the standard basis of $\mathbb{R}^{[n]}$. Let $A_n$ be random $n\times n$ submatrix of $B_n$, where the probability that we choose a submatrix $C$ is proportional to $|\det(C)|^2$. Let $p\ge 5$ be a prime. We prove that the asymptotic distribution of the $p$-Sylow subgroup of the cokernel of $A_n$ is given by the Cohen-Lenstra heuristics. Our result is motivated by the conjecture that the first homology group of a random two dimensional hypertree is also Cohen-Lenstra distributed.