  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.
