Asymptotic Problems in the Theory of Random Spectra

PI将研究有关随机矩阵和随机Schroedinger算子的谱的几个问题。 最重要的是，PI将继续他的研究随机矩阵没有对称性，建立波动结果的线性谱统计，极限定理的谱边缘，并进一步调查之间的联系，这些集合和根的随机多项式。在Hermitian随机矩阵的领域中，PI将使用Riemann-Hilbert问题方法来研究高维“酉系综”的Janossy密度的行为。 我们的目标是然后使用这些结果来建立最大特征值分布的收敛速度，它的限制Tracy-Widom分布。 最后，PI将把他最近关于一维周期Schroedinger算子的基态本征值分布的结果推广到更广泛的一类随机势，在这个建议中所研究的随机矩阵模型在多元统计等不同的领域有重要的应用（主成分分析），理论物理（量子系统的能级和共振）和电子工程（信号处理和无线通信）。 另一方面，具有随机势的Schroedinger算子长期以来一直被研究为无序固体中的基本模型。 PI带来承担各种技术从ProbabilityandAnalysis研究的详细行为的基本类的随机矩阵和随机薛定谔算子在物理相关的limitingregimes。 这个建议的总体目标是通过渐近分析揭示这些模型的新的共性或普遍性质。

The PI will investigate several problems concerning the spectra of randommatrices and random Schroedinger operators. Foremost, the PI willcontinue his study of random matrices with no symmetry, establishingfluctuation results for linear spectral statistics, limit theorems for thespectral edge, and further investigating the connections between theseensembles and roots of random polynomials. In the realm of Hermitianrandom matrices, the PI will employ the Riemann-Hilbert Problem method toinvestigate the behavior of the Janossy densities for large dimensional'Unitary Ensembles'. The goal is to then use these results to establishthe speed of convergence of the largest eigenvalue distribution to itslimiting Tracy-Widom distribution. Finally, the PI will generalize hisrecent results on the distribution of the ground state eigenvalue of aone-dimensional periodic Schroedinger operator with White Noise potentialto a broader class of random potentials.The random matrix models under study in this proposal have importantapplications to such disparate areas as multivariate statistics (principalcomponent analysis), theoretical physics (energy levels and resonances ofquantum systems) and electrical engineering (signal processing andwireless communication). Schroedinger operators with random potential onthe other hand have long been studied as fundmental models in disorderedsolids. The PI brings to bear a variety of techniques from Probabilityand Analysis to study the detailed behavior of basic classes of randommatrices and random Schroedinger operators in physically relevant limitingregimes. An overall goal of this proposal is to uncover newcommonalities, or universal properties, of these models through asymptoticanalysis.