  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.
