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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.
|Translated title of the contribution||The lowest eigenvalue of Jacobi random matrix ensembles and Painleve VI|
|Pages (from-to)||405204 - 405230|
|Number of pages||27|
|Journal||Journal of Physics A: Mathematical and Theoretical|
|Volume||43, number 40|
|Publication status||Published - Oct 2010|