Sharp threshold for universality of cokernels of random matrices over finite fields
Jungin Lee
TL;DR
The paper resolves a sharp universality threshold for cokernels of random matrices over finite fields, showing that for any fixed $c>1$ and $\alpha_n=\frac{c\log n}{n}$, the cokernel of an $n\times n$ matrix with $\alpha_n$-balanced entries in $\mathbb{F}_p$ converges in distribution to the same universal law as uniform random matrices, independent of the detailed entry distribution. The approach hinges on reducing the problem to extreme-point distributions, encoding cokernel moments via a Fourier-analytic framework with functions $f_{p,r}$ and $F_{p,r}$, and proving $F(t)\to 0$ by a meticulous decomposition of index ranges and casework (Cases C1–C3). The argument combines discrete Fourier analysis, combinatorial counting, and entropy-type estimates to bound contributions from all regimes, including large $i,j$ and structured subspace configurations, ultimately establishing near-optimal universality at the conjectured threshold. This result closes Woo22’s finite-field question and advances the broader universality program for cokernels of random $p$-adic and finite-field matrices, with implications for random Abelian $p$-groups and related combinatorial-probabilistic structures.
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 $A(n)$ 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).