Universal objects of the infinite beta random matrix theory

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.