Fluctuations of random matrix eigenvalues and disordered systems

这项NSERC发现补助金主要研究随机矩阵和随机增长模型的特征值的波动。经典的中心极限定理指出，无论单个随机变量的分布如何，大的独立随机变量和的分布都近似为正态分布。也就是说，当系统大小(随机变量的数量)趋于无穷大时，总和收敛到一个不依赖于系统微观细节的普遍概率分布。今天，概率论正在发现并试图理解一种新的普适性形式，这种普适性的特征是大的相关系统而不是独立的系统。普适性最早的形式之一是由物理学家维格纳提出的，他假设随机矩阵的本征值是描述大型量子系统(如重原子)能级的良好模型。自从Wigner的最初工作以来，随机矩阵统计已经被发现或被猜测出现在许多看似无关的环境中，包括在Riemann Zeta函数的零点中，在混沌经典系统的量子化中，以及在Kardar-Parisi-Zhang普适类中的随机增长模型中。这项研究有三个具体的研究目的，将通过以下三个方面加深我们对普适性现象的理解：(1)研究可解随机矩阵模型的特征多项式的极大值的行为，并进一步证明一般矩阵以及其他极值统计的普适性。(2)研究新环境下的特征值动力学，包括复平面上的动力学和酉阵上的动力学。(3)通过引入随机矩阵理论的思想和分析方法来研究一般的随机增长模型，而不是依赖于精确的公式，提出了随机矩阵特征值作为Riemann Zeta函数零点的概率模型，特征多项式的极大值和其他极值特征值统计量的研究部分是通过研究这种对应关系来实现的(研究目标1)。随机本征值动力学已经成为理解随机矩阵本征值的重要工具，并提供了随机矩阵和相互作用的粒子系统之间的联系(研究目标2)。一大类随机增长模型(KPZ普适性类)被猜想在大系统规模的极限下具有普适性。大多数可以证明这种行为的模型都是可以用精确公式处理的特例，这远远达不到最广泛的普适性愿景(研究目标3)。拟议研究的预期结果将是开发新的技术和方法，将增加我们对相关系统的普遍性的理解，并进一步建立随机矩阵理论与数学和科学的其他领域之间的联系。

This NSERC Discovery Grant is focused on studying fluctuations of eigenvalues of random matrices and random growth models. The classical central limit theorem states that the distribution of large sums of independent random variables looks approximately Gaussian, regardless of the distribution of the individual random variables. That is, as the system size (the number of random variables) tends to infinity, the sum converges to a universal probability distribution that does not depend on the microscopic details of the system. Today, probability theory is now discovering and trying to understand a new form of universality, that characterizes large correlated systems instead of independent ones. One of the first forms of universality was put forward by the physicist Wigner, who postulated that the eigenvalues of random matrices were a good model for the energy levels of large quantum systems, such as heavy atoms. Since Wigner's original work, random matrix statistics have been discovered or are conjectured to appear in a wide range of seemingly unrelated settings, including in the zeros of the Riemann zeta function, in the quantization of chaotic classical systems, and in random growth models in the Kardar-Parisi-Zhang universality class. The proposed research has 3 specific Research Aims that will further our understanding of this phenomena of universality by: (1) Investigating the behaviour of the maximum of the characteristic polynomial of solvable random matrix models and further proving universality for general matrices, as well as other extremal statistics. (2) Studying eigenvalue dynamics in new settings, including dynamics on the complex plane as well as those of unitary matrices. (3) Studying general random growth models by importing the ideas and analytic methods of random matrix theory, rather than relying on exact formulas.  Random matrix eigenvalues have been proposed as a probabilistic model for the zeros of the Riemann zeta function, and the study of the maximum of the characteristic polynomial and other extremal eigenvalue statistics is motivated in part by investigating this correspondence (Research Aim 1). Stochastic eigenvalue dynamics have emerged as an important tool for understanding random matrix eigenvalues, and provide a link between random matrices and interacting particle systems (Research Aim 2). Large classes of random growth models (the KPZ universality class) are conjectured to enjoy a universal behaviour in the limit of large system size.  The majority of models for which the behaviour can be proven are special cases that can be treated by exact formulas, which falls far short of the broadest visions of universality (Research Aim 3). The expected outcome of the proposed research will be to develop new techniques and methods that will increase our understanding of the universality of correlated systems, and further establish the links between random matrix theory and other areas of mathematics and science.