Soft Riemann-Hilbert problems and planar orthogonal polynomials

Its and Takhtajan studied a Riemann-Hilbert problem to investigate the asymptotics of orthogonal polynomials for exponentially varying weights in the complex plane. Later, using a foliation method, Hedenmalm and Wennman obtained an asymptotic expansion formula for those planar orthogonal polynomials (Acta Math, to appear). The foliation method was very complicated and hard to replicate in other situations. Here, we return to the original setting of Its and Takhtajan and show how to derive an even stronger asymptotic formula. We avoid the foliation, and replace it with a clever ansatz for the orthogonal polynomial and the associated potential, which are involved in the Riemann-Hilbert problem. This makes possible analysis of other situations.