Freezing limits for general random matrix ensembles and applications to classical β-ensembles
Loading...
Date
2024
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
This thesis studies the asymptotic behavior of eigenvalue distributions in random matrix theory, more specifically that of beta ensembles as the parameter beta tends to infinity. This regime is known as the “freezing limit”. Beta ensembles are often used to describe systems of interacting particles that are influenced by repulsive forces and external potentials. They appear in various physical and mathematical contexts.
In physics, beta ensembles play a key role in Calogero-Moser-Sutherland models. Here, beta represents the inverse temperature, and the freezing limit corresponds to rigid particle configurations. In mathematics, beta ensembles describe eigenvalue densities in classical random matrix models. Examples of such models that are studied in this thesis are the Gaussian, Wishart, and MANOVA ensemble. In the classical sense, the value of beta is fixed and depends on the underlying number field. For arbitrary positive beta, a tridiagonal matrix model can be defined such that its eigenvalue distribution is given by the beta ensemble for this beta.
The thesis focuses first on analyzing the behavior of beta ensembles for general convex potential functions in the freezing limit. These include the classical ensembles, the beta-Hermite, the beta-Laguerre the and beta-Jacobi ensemble as well as certain edge cases. Afterwards, the interplay of the freezing regime and pushing the number of particles to infinity is investigated in each of the three classical cases. These investigations provide new insights into the structural connection between beta ensembles and dual orthogonal polynomial systems.
Description
Table of contents
Keywords
Calogero-Moser-Sutherland method, Random matrices, β-ensembles, β-Hermite ensembles, β-Laguerre ensembles, β-Jacobi ensembles, Covariance matrices, Zeros of classical orthogonal polynomials, Dual orthogonal polynomials, Airy function, Bessel function