Traces of powers of matrices over finite fields

TL;DR

This paper establishes that traces of powers of random matrices over finite fields become uniformly distributed as matrix size grows, with exponential convergence rates, extending classical results from complex unitary groups to finite fields.

Contribution

It proves finite field analogues of trace distribution convergence and rate results, including for various classical groups, using techniques from analytic number theory over function fields.

Findings

Traces of matrix powers become uniformly distributed over finite fields as size increases.

Convergence rate is exponential in the square of the matrix size.

Results extend to classical groups like linear, symplectic, and orthogonal groups.

Abstract

Let $M$ be a random matrix chosen according to Haar measure from the unitary group $\mathrm{U}(n,\mathbb{C})$. Diaconis and Shahshahani proved that the traces of $M,M^2,\ldots,M^k$ converge in distribution to independent normal variables as $n \to \infty$, and Johansson proved that the rate of convergence is superexponential in $n$. We prove a finite field analogue of these results. Fixing a prime power $q = p^r$, we choose a matrix $M$ uniformly from the finite unitary group $\mathrm{U}(n,q)\subseteq \mathrm{GL}(n,q^2)$ and show that the traces of $\{ M^i \}_{1 \le i \le k,\, p \nmid i}$ converge to independent uniform variables in $\mathbb{F}_{q^2}$ as $n \to \infty$. Moreover we show the rate of convergence is exponential in $n^2$. We also consider the closely related problem of the rate at which characteristic polynomial of $M$ equidistributes in `short intervals' of $\mathbb{F}_{q^2}[T]$. Analogous results are also proved for the general linear, special linear, symplectic and orthogonal groups over a finite field. In the two latter families we restrict to odd characteristic. The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial.