Universality of the Stochastic Airy Operator

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.