Asymptotic Problems in the Theory of Random Spectra

PI将研究与随机矩阵和随机薛定谔算子的谱有关的几个问题。最重要的是，PI将继续他对不对称的随机矩阵的研究，建立线性谱统计的涨落结果，谱边的极限定理，并进一步研究这些系综与随机多项式的根之间的联系。在Hermite随机矩阵领域，PI将使用Riemann-Hilbert问题方法来研究大维“酉系综”的Janossy密度的行为。目标是利用这些结果来确定最大特征值分布到其极限Tracy-Widom分布的收敛速度。最后，PI将把他最近关于具有白噪声势的一维周期薛定谔算子基态本征值分布的结果推广到更广泛的随机势。本文中研究的随机矩阵模型在多元统计(主成分分析)、理论物理(量子系统的能级和共振)和电气工程(信号处理和无线通信)等不同领域有重要的应用。另一方面，具有随机势的薛定谔算符长期以来一直被作为无序固体中的基本模型来研究。PI运用概率与分析中的各种技术来研究随机矩阵和随机薛定谔算子的基本类在物理上相关的极限区域中的详细行为。这一建议的一个总体目标是通过渐近分析来发现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.