The Lowest Eigenvalue of Jacobi Random Matrix Ensembles and Painlevé VI

Dueñez, E.; Huynh, D. K.; Keating, J. P.; Miller, S. J.; Snaith, N. C.

J. Phys. A 43 (2010), no. 40, 405204, 27 pp.

“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 Painlevé VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the Painlevé 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.”