Universal objects of the infinite beta random matrix theory

We develop a theory of multilevel distributions of eigenvalues which complements Dyson's threefold 13 = 1, 2, 4 approach corresponding to real/complex/quaternion matrices by 13 = oo point. Our central objects are the GooE ensemble, which is a counterpart of the classical Gaussian Orthogonal/Unitary/Symplectic ensembles, and the Airy1 line ensemble, which is a collection of continuous curves serving as a scaling limit for largest eigenvalues at 13 = oo. 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.