Traces of powers of matrices over finite fields

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.