Universal objects of the infinite beta random matrix theory

  • Vadim Gorin

    University of California, Berkeley, USA
  • Victor Kleptsyn

    Université de Rennes, France
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Abstract

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

Cite this article

Vadim Gorin, Victor Kleptsyn, Universal objects of the infinite beta random matrix theory. J. Eur. Math. Soc. 26 (2024), no. 9, pp. 3429–3496

DOI 10.4171/JEMS/1336