Sharp threshold for universality of cokernels of random matrices over finite fields

In this paper, we determine the sharp threshold for universality of cokernels of random matrices over finite fields. More precisely, we prove the following: given any constant $c>1$, let $A(n)$ be a random $n \times n$ matrix over $\mathbb{F}_p$ whose entries are independent and take any given value of $\mathbb{F}_p$ with probability at most $1 - \frac{c \log n}{n}$. Then the cokernels of $A(n)$ converge in distribution, as $n \to \infty$, to the same limiting law as the cokernels of uniform random $n \times n$ matrices over $\mathbb{F}_p$. This answers an open problem posed by Wood (2022).