Spectral and Hierarchical Properties of Random Matrices

随机矩阵的谱统计描述了许多无序物理系统的能级（本征态）的关键特性。其应用范围最近有所扩大。例如，可积增长模型、量子力学和解析数论表现出随机矩阵特征值分布。随机矩阵的行列式也是分析许多高维随机函数的工具。应用包括深度学习中损失函数优化可能性的标准。最近，出现了一种新的联系，即随机矩阵和L函数的特征值背后的层次结构，它们的极值的特征在于与分支过程的类比。这个项目将扩展和联合收割机最近的方法开发这些随机矩阵谱和层次的普遍性，扩大其范围，并将这些技术应用到随机景观。该项目为研究生提供了研究培训的机会。PI将首先研究随机矩阵，分支过程，L-函数和高斯乘性混沌之间的普遍联系，通过研究Fyodorov-Keating矩阵的最大紧度，它们在Wigner矩阵和Beta系综类中的普遍性，从随机正规矩阵的特征多项式出发，证明了二维高斯乘性混沌测度的出现。PI将Fisher-Hartwig渐近公式扩展到允许任何温度的设置。PI还将研究随机景观的拓扑复杂性及其临界指数的普适性，因为信号和噪声变化。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The spectral statistics of random matrices describe key properties of the energy levels (eigenstates) of many disordered physical systems. The scope of their applications has expanded recently. For example, integrable growth models, quantum mechanics, and analytic number theory exhibit random matrix eigenvalue distributions. Determinants of random matrices also serve as tools for the analysis of many high-dimensional random functions. Applications include criteria for the possibility of optimization of loss functions in deep learning. More recently, a new connection has emerged, a hierarchical structure behind the eigenvalues of random matrices and L-functions, their extremes being characterized by analogy with branching processes. This project will extend and combine recent methods developed for these random matrix spectral and hierarchical universalities to enlarge their scope, and apply these techniques to random landscapes. The project provides research training opportunities for graduate students.The PI will first work on universal connections between random matrices, branching processes, L-functions and gaussian multiplicative chaos, by studying the Fyodorov-Keating conjectures up to tightness of the maximum, their universality in the class of Wigner matrices and beta ensembles, and the emergence of the 2d Gaussian multiplicative chaos measures from the characteristic polynomial of random normal matrices. The PI will extend the Fisher-Hartwig asymptotic formulas to settings allowing any temperature. The PI will also study the topological complexity of random landscapes and the universality of their critical exponents, as the signal and noise vary.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.