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

TL;DR

This paper analyzes the asymptotic behavior of orthogonal polynomials with complex quartic weights, identifying critical points, and describing the local behavior near these points using advanced Riemann-Hilbert techniques.

Contribution

It provides a comprehensive global analysis of recurrence coefficients for complex quartic weights, discovering new critical points and detailing asymptotics near these points with error estimates.

Findings

Identified all critical points, including new ones at t_1=1/15 and t_2=1/4.

Derived asymptotics of recurrence coefficients near critical points, including spike shapes.

Developed a universal description of spike behavior at poles of Painleve' I solutions.

Abstract

We study the asymptotics of recurrence coefficients for monic orthogonal polynomials p_n(z) with the quartic exponential weight exp [-N (1/2 z^2 + t/4 z^4)], 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 t_0=- 1/ 12, we found new critical points t_1=1/15 and t_2=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 t_0,t_1 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.