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arXiv · 1005.1298

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

Eduardo Dueñez·Duc Khiem Huynh·Jon P. Keating·Steven J. Miller·Nina C. Snaith
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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.

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Eduardo Dueñez, Duc Khiem Huynh, Jon P. Keating, Steven J. Miller, Nina C. Snaith. 2010-05-07. The lowest eigenvalue of Jacobi random matrix ensembles and Painlev\'e VI. https://doi.org/10.1088/1751-8113/43/40/405204

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