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

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.