Boundary conditions associated with the Painlevé III' and V evaluations of some random matrix averages

Abstract

In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval $(0,s)$ at the hard edge contains $k$ eigenvalues, was evaluated in terms of a Painlevé V transcendent in $\sigma$-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small $s$ behaviour of the Painlevé V equation in $\sigma$-form known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlevé \IIId transcendent in $\sigma$-form. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function.

References in corpus (1)

• Calculation of some determinants using the s-shifted factorial

Cited by in corpus (12)

• A matrix model with a singular weight and Painleve' III
• Mixed moments of characteristic polynomials of random unitary matrices
• Random Matrix Theory and the Sixth Painlevé Equation
• A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions
• Painlevé III${}^{\prime }$ and the Hankel Determinant Generated by a Singularly Perturbed Gaussian Weight
• Perturbed Hankel determinant, correlation functions and Painlevé equations
• Painlevé V and the Hankel Determinant for a Singularly Perturbed Jacobi Weight
• Joint moments of a characteristic polynomial and its derivative for the circular $\beta$-ensemble
• Convergence and an explicit formula for the joint moments of the Circular Jacobi $\beta$-Ensemble characteristic polynomial
• Joint moments of derivatives of characteristic polynomials of random symplectic and orthogonal matrices
• Boundary Conditions for Scaled Random Matrix Ensembles in the Bulk of the Spectrum
• Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters