Moments of traces of random symplectic matrices and hyperelliptic $L$-functions

Abstract

We study matrix integrals of the form $${\int }_{\mathrm{USp}(2n)}\prod _{j=1}^k\mathrm{tr}(U^j{)}^{a_j}\mathrm{d}U,$$ where $a_1,\dots ,a_r$ are natural numbers and integration is with respect to the Haar probability measure. We obtain a compact formula (the number of terms depends only on $\sum a_j$ and not on $n,k$) for the above integral in the non-Gaussian range ${\sum }_{j=1}^{k}j{a}_j\le 4n+1$. This extends results of Diaconis-Shahshahani and Hughes-Rudnick who obtained a formula for the integral valid in the (Gaussian) range ${\sum }_{j=1}^{k}j{a}_j\le n$ and ${\sum }_{j=1}^{k}j{a}_j\le 2n+1$ respectively. We derive our formula using the connection between random symplectic matrices and hyperelliptic $L$-functions over finite fields, given by an equidistribution result of Katz and Sarnak, and an evaluation of a certain multiple character sum over the function field ${\mathbb{F}}_q(x)$. We apply our formula to study the linear statistics of eigenvalues of random unitary symplectic matrices in a narrow bandwidth sampling regime.

v2: minor corrections and slightly improved exposition