Abstract: In this paper, we study the asymptotics of polynomials orthogonal with respect to the varying quartic weight ω(x) = e−nV(x), where 
. 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).
