Universal objects of the infinite beta random matrix theory

Abstract

We develop a theory of multilevel distributions of eigenvalues which complements Dyson’s threefold $\beta =1,2,4$ approach corresponding to real/complex/quaternion matrices by $\beta =\infty$ point. Our central objects are the G$\infty$E ensemble, which is a counterpart of the classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and the 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 view on these objects. The probabilistic one treats them as partition functions of certain additive polymers collecting white noise. The integrable point of view expresses their distributions through the so-called associated Hermite polynomials and integrals of the Airy function. We also outline universal appearances of our ensembles as scaling limits.