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The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI. (English) Zbl 1206.34113

Summary: 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.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
60B20 Random matrices (probabilistic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
15B52 Random matrices (algebraic aspects)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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