Global asymptotics for polynomials orthogonal with exponential quartic weight

Abstract

In this paper, we study the asymptotics of polynomials orthogonal with respect to the varying quartic weight ω(x)=e^−nV(x), where V(x)=V_t(x)=x⁴/4+x²t/2. We focus on the critical case t=−2, in the sense that for t≥−2, the support of the associated equilibrium measure is a single interval, while for t<−2, the support consists of two intervals. Globally uniform asymptotic expansions are obtained for z in three unbounded regions. These regions together cover the whole complex z-plane. In particular, in the region containing the origin, the expansion involves the Ψ function affiliated with the Hastings–McLeod solution of the second Painlevé equation. Our approach is based on a modified version of the steepest-descent method for Riemann–Hilbert problems introduced by Deift and Zhou (Ann. Math. 137 (1993), 295–370).