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

TL;DR

This paper calculates the probability that a matrix from the gGUE is positive definite by analyzing the asymptotics of semi-classical Laguerre polynomials through Riemann-Hilbert methods.

Contribution

It extends previous results by deriving the large degree asymptotics of semi-classical Laguerre polynomials for the gGUE.

Findings

Derived asymptotics of semi-classical Laguerre polynomials

Computed positive-definiteness probability for gGUE matrices

Extended prior results of Dean and Majumdar

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.