  We obtain Plancherel-Rotach type asymptotics valid in all regions of the
complex plane for orthogonal polynomials with varying weights of the form
$e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz
continuous derivatives and that the corresponding equilibrium measure has
typical support properties. As an application we extend the universality class
for bulk and edge asymptotics of eigenvalue statistics in unitary invariant
Hermitian random matrix theory. Our methodology involves developing a new
technique of asymptotic analysis for matrix Riemann-Hilbert problems with
nonanalytic jump matrices suitable for analyzing such problems even near
transition points where the solution changes from oscillatory to exponential
behavior.
