Universality of the Stochastic Airy Operator

TL;DR

This paper introduces a new method demonstrating that the scaled Jacobi matrices from Dyson beta ensembles converge to the Stochastic Airy operator, establishing universality at the spectral edge.

Contribution

It presents a novel approach for proving the universality of the soft edge in random matrix ensembles through operator convergence.

Findings

Convergence of scaled Jacobi matrices to the Stochastic Airy operator.

Universality of the top eigenvalue distribution in Dyson beta ensembles.

Conjectured operator limits for all soft edge distributions.

Abstract

We introduce a new method for studying universality of random matrices. Let T_n be the Jacobi matrix associated to the Dyson beta ensemble with uniformly convex polynomial potential. We show that after scaling, T_n converges to the Stochastic Airy operator. In particular, the top edge of the Dyson beta ensemble and the corresponding eigenvectors are universal. As a byproduct, our work leads to conjectured operator limits for the entire family of soft edge distributions.