标题： 具有复数变化四次权的正交多项式的渐近性：全局结构、临界点行为及第一 Painlevé 方程 显示英文标题

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

摘要： 我们研究了具有四次指数权 exp[-N(1/2 z² + t/4 z⁴)] 的首一正交多项式 pₙ(z) 的递推系数的渐近性，其中 t 是复数。 我们的目标是：A) 在 t 中全局描述不同渐近行为（不同 genus）的区域；B) 确定所有临界点；C) 详细研究临界点附近全邻域内的渐近性（双尺度极限），包括 Painlevé I 方程解 y(v) 的极点附近和附近，已知这些极点在此极限下提供主要修正项。 我们的结果是：A) 我们找到了不同积分路径配置下正交多项式的递推系数和“平方范数”的全局（关于 t）渐近性。专门开发了代码来分析所有可能的情况。 B) 除了已知的临界点 t₀ = -1/12 外，我们还发现了新的临界点 t₁ = 1/15 和 t₂ = 1/4。 C) 我们导出了在 y(v) 的极点以及 t₀、t₁ 附近的临界点处及其周围的三重尺度极限下的递推系数的主要阶行为（连同误差估计）。我们证明了递推系数在 y(v) 的极点附近有无界的“尖峰”，并计算了不同情况下的“普适”形状。 本文的主要技术是 Riemann-Hilbert 问题（RHP）的非线性最速下降法。我们注意到，临界点附近的 RHP 与作者之前解决的聚焦 NLS 在梯度灾难点附近的半经典极限描述的 RHP 非常相似。 显示英文摘要

摘要： 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.