Article

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

05/2010; DOI:abs/1005.1298
Source: arXiv

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. Comment: 30 pages, 2 figures

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Article:Painlev\'e VI and Hankel determinants for the generalized Jacobi Weight
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Article:Random matrix theory and the sixth Painlevé equation
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Keywords

2 figures

associated Painleve VI nonlinear differential equation numerically

central point

elliptic curve L-function families

finite conductor

first method solves

Painleve VI function

random matrices

suitable initial conditions