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We study a family of monic orthogonal polynomials which are orthogonal withrespect to the varying, complex valued weight function, $\exp(nsz)$, over theinterval $[-1,1]$, where $s\in\mathbb{C}$ is arbitrary. This family ofpolynomials originally appeared in the literature when the parameter was purelyimaginary, that is $s\in i \mathbb{R}$, due to its connection with complexGaussian quadrature rules for highly oscillatory integrals. The asymptotics forthese polynomials as $n\to\infty$ have been recently studied for $s\ini\mathbb{R}$, and our main goal is to extend these results to all $s$ in thecomplex plane.  We first use the technique of continuation in parameter space, developed inthe context of the theory of integrable systems, to extend previous results onthe so-called modified external field from the imaginary axis to the complexplane minus a set of critical curves, called breaking curves. We then apply thepowerful method of nonlinear steepest descent for oscillatory Riemann-Hilbertproblems developed by Deift and Zhou in the 1990s to obtain asymptotics of therecurrence coefficients of these polynomials when the parameter $s$ is awayfrom the breaking curves. We then provide the analysis of the recurrencecoefficients when the parameter $s$ approaches a breaking curve, by consideringdouble scaling limits as $s$ approaches these points. We shall see aqualitative difference in the behavior of the recurrence coefficients,depending on whether or not we are approaching the points $s=\pm 2$ or someother points on the breaking curve.

Global Phase Portrait and Large Degree Asymptotics for the Kissing Polynomials