We study whether in the setting of the Deift–Zhou nonlinear steepest descent method one can avoid solving local parametrix problems, while still obtaining asymptotic results. We show that this can be done, provided an a priori estimate for the exact solution of the Riemann–Hilbert problem is known. This enables us to derive asymptotic results for orthogonal polynomials on $[-1,1]$ with a new class of weight functions. In these cases, the weight functions are too badly behaved to allow a reformulation of the local parametrix problem to a global one with constant jump matrices. Possible implications for edge universality in random matrix theory are also discussed.

我们研究在 Deift-Zhou 非线性最速下降法的设置中是否可以避免解决局部参数问题，同时仍然获得渐近结果。我们表明这是可以做到的，前提是已知黎曼-希尔伯特问题的精确解的先验估计。这使我们能够推导出正交多项式的渐近结果$[-1,1]$使用一类新的权重函数。在这些情况下，权重函数的表现太差，无法将局部参数问题重新表述为具有恒定跳跃矩阵的全局问题。还讨论了随机矩阵理论中边缘普遍性的可能含义。