### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 118, 50 pages      arXiv:1612.08732      https://doi.org/10.3842/SIGMA.2016.118
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

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

Marco Bertola a and Alexander Tovbis b
a) Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, Canada H3G 1M8
b) University of Central Florida, Department of Mathematics, 4000 Central Florida Blvd., P.O. Box 161364, Orlando, FL 32816-1364, USA

Received March 11, 2016, in final form December 12, 2016; Published online December 27, 2016

Abstract
We study the asymptotics of recurrence coefficients for monic orthogonal polynomials $\pi_n(z)$ with the quartic exponential weight $\exp [-N (\frac 12 z^2 + \frac 14 t z^4)]$, 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'' curves separating these regions, and discuss their special (critical) points. All these pieces of information combined provide the global asymptotic ''phase portrait'' of the recurrence coefficients of $\pi_n(z)$, which was studied numerically in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321]. The main goal of the present paper is to provide a rigorous framework for the global asymptotic portrait through the nonlinear steepest descent analysis (with the $g$-function mechanism) of the corresponding Riemann-Hilbert problem (RHP) and the continuation in the parameter space principle. The latter allows to extend the nonlinear steepest descent analysis from some parts of the complex $t$-plane to all noncritical values of $t$. We also provide explicit solutions for recurrence coefficients in terms of the Riemann theta functions. The leading order behaviour of the recurrence coefficients in the full scaling neighbourhoods the critical points (double and triple scaling limits) was obtained in [Constr. Approx. 41 (2015), 529-587, arXiv:1108.0321] and [Asymptotics of complex orthogonal polynomials on the cross with varying quartic weight: critical point behaviour and the second Painlevé transcendents, in preparation].

Key words: double scaling limit of orthogonal polynomials; asymptotics of recurrence coefficients; method of Riemann-Hilbert problem; nonlinear steepest descent analysis.

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