Distribution of the cokernels of determinantal row-sparse matrices

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. Mészáros which established convergence to the Cohen–Lenstra distribution when $p\ge 5$ and $k_n=3$ for all positive integers $n$.