The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI

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.

30 pages, 2 figures

References in corpus (3)

• Painlevé VI and Hankel determinants for the generalized Jacobi Weight
• Random Matrix Theory and the Sixth Painlevé Equation
• Painlevé VI and the Unitary Jacobi ensembles

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• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"