Global asymptotics for polynomials orthogonal with exponential quartic weight

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 + t/2 x². 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). © 2009 - IOS Press and the authors. All rights reserved.