Abstract: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 $(A(n))_{n \ge 1}$ be a sequence of random $n \times n$ matrices over $\mathbb{F}_p$ such that, for all sufficiently large $n$, the entries of $A(n)$ 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.
From: Jungin Lee [view email]
[v1]
Mon, 17 Nov 2025 07:14:58 UTC (14 KB)
[v2]
Sun, 23 Aug 2026 12:32:16 UTC (18 KB)
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