On the probability of positive-definiteness in the gGUE via semi-classical Laguerre polynomials

Abstract

In this paper, we compute the probability that an $N\times N$ matrix from the generalised Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar \cite{DM}. For this purpose, we work out the large degree asymptotics of semi-classical Laguerre polynomials and their recurrence coefficients, using the steepest descent analysis of the corresponding Riemann--Hilbert problem.

21 pages, 1 figure. Revised version, minor changes and references added

References in corpus (5)

• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Cosmology From Random Multifield Potentials
• Lectures on random matrix models. The Riemann-Hilbert approach
• What is the probability that a random integral quadratic form in $n$ variables has an integral zero?
• Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble

Cited by in corpus (4)

• Asymptotics of Hankel determinants with a one-cut regular potential and Fisher-Hartwig singularities
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• Painlevé IV, Chazy II, and Asymptotics for Recurrence Coefficients of Semi-classical Laguerre Polynomials and Their Hankel Determinants
• Asymptotics for Hankel Determinants Associated to a Hermite Weight with a Varying Discontinuity