Cohen–Lenstra Distribution for Sparse Matrices with Determinantal Biasing

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.