Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painleve' equation

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We study the asymptotics of recurrence coefficients for monic orthogonal polynomials pₙ(z) with the quartic exponential weight exp [-N (1/2 z² + t/4 z⁴)], where t is complex. Our goals are: A) to describe the regions of different asymptotic behaviour (different genera) globally in t; B) to identify all the critical points, and; C) to study in details the asymptotics in a full neighborhood near of critical points (double scaling limit), including at and near the poles of Painleve' I solutions y(v) that are known to provide the leading correction term in this limit. Our results are: A) We found global (in t) asymptotic of recurrence coefficients and of "square-norms" for the orthogonal polynomials for different configurations of the contours of integration. Special code was developed to analyze all possible cases. B) In addition to the known critical point t₀=- 1/ 12, we found new critical points t₁=1/15 and t₂=1/4. C) We derived the leading order behavior of the recurrence coefficients (together with the error estimates) at and around the poles of y(v) near the critical points t₀,t₁ in what we called the triple scaling limit. We proved that the recurrence coefficients have unbounded "spikes" near the poles of y(v) and calculated the "universal" shape of these spikes for different cases. The nonlinear steepest descent method for Riemann-Hilbert Problem (RHP) is the main technique used in the paper. We note that the RHP near the critical points is very similar to the RHP describing the semiclassical limit of the focusing NLS near the point of gradient catastrophe that the authors solved previously.