Universality of the Stochastic Airy Operator

Understand

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.