Singularity of Random Matrices over Finite Fields

TL;DR

This work studies the singularity probability and determinant distribution of an $n\times n$ random matrix over the finite field $\mathbb{F}_q$ with iid entries from an $\alpha$-dense distribution. It introduces a universality framework by reducing to a Littlewood–Offord problem over finite fields and classifying potential subspaces into sparse, semi-saturated, and unsaturated types, then proving three key Littelwood–Offord results (classical LO, inverse theorem, swapping lemma) to achieve exponential-rate bounds. The main contributions are exact exponential-rate convergence: the singularity probability approaches the universal density $\prod_{k=1}^n (1 - q^{-k})$ with error $O(e^{-c\alpha n})$, and for any nonzero $t$, $\mathbb{P}(\det A = t) = q^{-1} \prod_{k=2}^ty (1 - q^{-k}) + O(e^{-c\alpha n})$, with analogous exponential control for the determinant distribution. These results establish robust universality for random matrices over finite fields under mild density assumptions and quantify the convergence with explicit exponential rates, enhancing understanding of spectral properties in noncontinuous, discrete settings.

Abstract

Let $A$ be an $n \times n$ random matrix with iid entries over a finite field of order $q$. Suppose that the entries do not take values in any additive coset of the field with probability greater than $1 - α$ for some fixed $0 < α< 1$. We show that the singularity probability converges to the uniform limit with an exponentially small error depending only on $α$. We also show that the distribution of the determinant of $A$ converges to its limiting distribution at an exponential rate.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Lemma 2.2: Odlyzko
• proof : Proof of Lemma \ref{['lem:odlyzko']}
• Proposition 2.3
• proof
• Lemma 2.4: Littlewood-Offord
• Proposition 2.5
• Definition 2.6