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

2024/09/07 by Alexei Entin, Entin, Alexei, Noam Pirani +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2409.04844

openalex publication_date 2024/09/07 · openalex created_date 2024/10/22 · openalex updated_date 2026/07/28

Abstract

We study matrix integrals of the form ∫_USp(2n)∏_j=1^ktr(U^j)^a_j\mathrm d U, where a₁,…,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 ∑ a_j and not on n,k) for the above integral in the non-Gaussian range ∑_j=1^kja_j≤ 4n+1. This extends results of Diaconis-Shahshahani and Hughes-Rudnick who obtained a formula for the integral valid in the (Gaussian) range ∑_j=1^kja_j≤ n and ∑_j=1^kja_j≤ 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.

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