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We study the asymptotics of recurrence coefficients for monic orthogonalpolynomials $\pi_n(z)$ with the quartic exponential weight $\exp[-N(\frac 12z^2+\frac 14 tz^4)]$, where $t\in {\mathbb C}$ and $N\in{\mathbb N}$,$N\to\infty$. Our goal is to describe these asymptotic behaviors globally for$t\in {\mathbb C}$ in different regions. We also describe the "breaking" curvesseparating these regions, and discuss their special (critical) points. Allthese pieces of information combined provide the global asymptotic "phaseportrait" of the recurrence coefficients of $\pi_n(z)$, which was studiednumerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The maingoal of the present paper is to provide a rigorous framework for the globalasymptotic portrait through the nonlinear steepest descent analysis (with the$g$-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) andthe continuation in the parameter space principle. The latter allows to extendthe nonlinear steepest descent analysis from some parts of the complex$t$-plane to all noncritical values of $t$. We also provide explicit solutionsfor recurrence coefficients in terms of the Riemann theta functions. Theleading order behaviour of the recurrence coefficients in the full scalingneighbourhoods the critical points (double and triple scaling limits) wasobtained in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321] and[Asymptotics of complex orthogonal polynomials on the cross with varyingquartic weight: critical point behaviour and the second Painlev\'etranscendents, in preparation].

On Asymptotic Regimes of Orthogonal Polynomials with Complex Varying Quartic Exponential Weight