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

2016/10/31 by Alfredo Deaño, Nick Simm, Nicholas J. Simm
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebra over a field #Definiteness #Eigenvalues and eigenvectors #Laguerre polynomials #Linguistics #Mathematical functions and polynomials #Mathematics #Positive definiteness #Positive-definite matrix #Pure mathematics #Random Matrices and Applications #math-ph #math.CA #math.MP #msc:33C45 #msc:34E05 #msc:60B20

paper · pdf · doi:10.1016/j.jat.2017.04.004

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

openalex publication_date 2017/05/06 · arxiv created 2017/05/23 · arxiv updated 2017/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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

Citations

• Classical and quantum orthogonal polynomials in one variable
• The isomonodromy approach to matric models in 2D quantum gravity
• Lectures on random matrix models. The Riemann-Hilbert approach
• Glassy random matrix models
• Cosmology from random multifield potentials
• Density of Positive Eigenvalues of the Generalized Gaussian Unitary Ensemble
• What is the probability that a random integral quadratic form in n variables has an integral zero?
• Classical and Quantum Orthogonal Polynomials in One Variable