Universal objects of the infinite beta random matrix theory

TL;DR

This paper introduces a new framework for understanding eigenvalue distributions in infinite beta random matrix theory, extending classical ensembles to the infinite beta case with new universal objects.

Contribution

It develops the G∞E ensemble and Airy∞ line ensemble as infinite beta analogs of classical matrix ensembles, providing probabilistic and integrable perspectives.

Findings

Defined G∞E ensemble as a counterpart to classical ensembles

Introduced Airy∞ line ensemble as a scaling limit for largest eigenvalues

Outlined universality of these ensembles as scaling limits

Abstract

We develop a theory of multilevel distributions of eigenvalues which complements the Dyson's threefold $\beta=1,2,4$ approach corresponding to real/complex/quaternion matrices by $\beta=\infty$ point. Our central objects are G$\infty$E ensemble, which is a counterpart of classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and Airy$_{\infty}$ line ensemble, which is a collection of continuous curves serving as a scaling limit for largest eigenvalues at $\beta=\infty$. We develop two points of views on these objects. Probabilistic one treats them as partition functions of certain additive polymers collecting white noise. Integrable point of view expresses their distributions through the so-called associated Hermite polynomials and integrals of Airy function. We also outline universal appearances of our ensembles as scaling limits.