Discrete Painlevé equations and random matrix averages

Abstract

The $\tau$-function theory of Painlevé systems is used to derive recurrences in the rank $n$ of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlevé equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The recurrences are illustrated by computing the value of a sequence of these distributions as $n$ varies, and demonstrating convergence to the value of the appropriate limiting distribution.

25 pages

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• Painlevé Transcendents and the Hankel Determinants Generated by a Discontinuous Gaussian Weight
• Discrete Painlevé equations for a class of \PVI $\tau$-functions given as U(N) averages
• Growth models, random matrices and Painleve transcendents
• Painleve versus Fuchs
• Tau-function of discrete isomonodromy transformations and probability
• Spherical spin-glass - Coulomb gas duality: solution beyond mean-field theory
• Level-spacing distributions of the Gaussian unitary random matrix ensemble
• On Integrable Ermakov-Painlevé IV Systems
• Restoring discrete Painlevé equations from an E${}_8^{(1)}$-associated one
• Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equation
• Moduli spaces of d-connections and difference Painleve equations
• Differential equations for deformed Laguerre polynomials
• Asymptotics of the partition function of the perturbed Gross-Witten-Wadia unitary matrix model