Universality for cokernels of partially random integral matrices

Given any $\varepsilon > 0$, let $M(n)$ be a random $n \times (n+u)$ matrix over $\mathbb{Z}_p$, with all entries independent and $\varepsilon$-balanced (lying in each residue class mod $p$ with probability at most $1-\varepsilon$). Wood proved that as $n \to \infty$ the distribution of $\mathrm{cok}(M(n))$ approaches Cohen and Lenstra's conjectured distribution of class groups. Given $\alpha,\beta >0$ such that $\alpha + \beta <1$, we prove that the distribution of $\mathrm{cok}(M(n))$ still approaches the Cohen--Lenstra distribution even if we weaken the hypothesis by allowing up to $\alpha n$ entries per column and up to $\beta n$ entries per row of $M(n)$ to not be $\varepsilon$-balanced. We also weaken the independence condition by allowing certain types of dependence between the entries of each column. In addition, we prove that, for any $\delta > 0$, the cokernels of random band matrices of width $\log(n)^{1+\delta}$ with $\varepsilon$-balanced entries in the band and arbitrary entries outside of it will also approach the Cohen--Lenstra distribution, which answers a question of Kang--Lee--Yu.