Characteristic Polynomials of Orthogonal and Symplectic Random Matrices, Jacobi Ensembles & L-functions

TL;DR

This paper explores the asymptotic behavior of characteristic polynomials of symplectic matrices and their derivatives, linking these results to conjectures about L-functions and introducing new computations in random matrix theory.

Contribution

It provides new asymptotic formulas for joint moments of characteristic polynomials and their derivatives, connecting these to Painleve equations and L-function conjectures.

Findings

Asymptotics of joint moments involving derivatives are derived.

Leading order coefficients are expressed via Painleve equations.

Results include asymptotics for additive Jacobi statistics.

Abstract

Starting from Montgomery's conjecture, there has been a substantial interest on the connections of random matrix theory and the theory of L-functions. In particular, moments of characteristic polynomials of random matrices have been considered in various works to estimate the asymptotics of moments of L-function families. In this paper, we first consider joint moments of the characteristic polynomial of a symplectic random matrix and its second derivative. We obtain the asymptotics, along with a representation of the leading order coefficient in terms of the solution of a Painleve equation. This gives us the conjectural asymptotics of the corresponding joint moments over families of Dirichlet L-functions. In doing so, we compute the asymptotics of a certain additive Jacobi statistic, which could be of independent interest in random matrix theory. Finally, we consider a slightly different type of joint moment that is the analogue of an average considered over the unitary group in various works before. We obtain the asymptotics and the leading order coefficient explicitly.