The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI

Details The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI

2010-05-07

arxiv.org/abs/1005.1298v2

Mathematics

Classical Analysis and ODEs

30 pages, 2 figures

Scientific paper

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.

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