Universality Limits at the Soft Edge of the Spectrum via Classical Complex Analysis

We show that universality at the soft edge of the spectrum is equivalent to universality “along the diagonal,” that is ratio asymptotics for Christoffel functions. The context is that of varying measures and limits involving the Airy kernel. In particular, we consider measures of the form $W^{2n}_n (x)\ {\mathrm d}x$, where {Wn} are a suitable sequence of weights. They do not need to be analytic, but instead should satisfy some hypotheses on the associated equilibrium measures.