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

In this paper, we compute the probability that an N×N matrix from the generalized Gaussian Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and Majumdar (2008). 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.