The lowest eigenvalue of Jacobi random matrix ensembles and Painlev\'e VI

TL;DR

This paper develops two methods to evaluate the distribution of the lowest eigenvalue in Jacobi random matrix ensembles using Painleve VI equations, with applications to elliptic curve L-functions.

Contribution

It introduces numerical and series expansion techniques for Painleve VI equations to analyze eigenvalue distributions in Jacobi ensembles, bridging random matrix theory and number theory.

Findings

Validated methods for Painleve VI-based eigenvalue distribution computation

Applied techniques to model zeros of elliptic curve L-functions

Provided tools for analyzing eigenvalues in finite-conductor matrix ensembles

Abstract

We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Painleve VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the Painleve VI function. Our results are applied in a forthcoming paper in which we model the distribution of the first zero above the central point of elliptic curve L-function families of finite conductor and of conjecturally orthogonal symmetry.