In this important and interesting paper, the authors again illustrate the awesome power of the Deift-Zhou steepest descent method associated with Riemann-Hilbert problems. Let be real-valued and analytic on , with
For , let denote the sequence of orthonormal polynomials for the weight , so that
The authors derive Plancherel-Rotach asymptotics for as , valid in every region of the plane. The precision of the asymptotics on and off the real line is remarkable. The proofs involve the Fokas-Its-Kitaev identify for the orthonormal polynomials as solutions of a Riemann-Hilbert problem, followed by the Deift-Zhou steepest descent technique. The details are intricate, but are clearly presented, and the main ideas are summarized to guide the reader through the proofs.
As an application, the authors prove universality limits that arise in random matrix theory. These involve the weighted reproducing kernel
and have the form
uniformly for , in compact subsets of . Here lies in a subinterval of the support of the equilibrium measure associated with the field , and is the density of that equilibrium measure.