Random Matrices and Interacting Systems

20世纪50年代，物理学家尤金·维格纳提出研究大型随机对称矩阵，以逼近复杂自伴算子的行为。他的目标是了解随机矩阵本征值的标度特性，以深入了解研究重核所产生的某些算符的光谱。在过去的半个世纪里，随机矩阵在数学(例如组合学、数论和相互作用的随机系统的研究)和各种其他领域(例如信息论、金融数学和RNA折叠)中得到了广泛的应用。近年来，利用某些随机自伴微分算子的谱，建立了随机矩阵极限律的一个新的刻画。这在Wigner最初的想法半个多世纪之后，在随机矩阵和自伴算子之间提供了一种新的联系。这个研究项目探索了这个新的领域。这个项目建立在研究者和合作者关于随机矩阵尺度极限的随机算子表示的最新结果的基础上。其目的是研究合成的随机算子，以便更多地了解极限点过程。所研究的问题包括将经典的微分算子理论应用于随机矩阵，研究具有定量误差界的随机矩阵模型的算子级收敛结果，以及研究随机矩阵极限与双曲平面中扩散之间的联系。该项目还包括与研究相互作用的随机系统的大规模行为有关的调查。推测一维相互作用的随机系统具有与随机矩阵理论有关的极限分布的不寻常的标度行为，这就是Kardar-Parisi-Zhang(KPZ)普适类。研究人员打算探索属于(或推测属于)KPZ普适性类的各种模型，特别关注定向聚合物，这是一种描述时空环境中的随机路径的模型，也是随机的。

In the 1950's the physicist Eugene Wigner proposed the study of large random symmetric matrices in order to approximate the behavior of complicated self-adjoint operators. His goal was to understand the scaling properties of the eigenvalues of random matrices to get insight about the spectrum of certain operators arising from the study of heavy nuclei. In the last half century, random matrices found applications in a wide range of areas both within mathematics (e.g., combinatorics, number theory, and the study of interacting stochastic systems) and various other fields as well (e.g., information theory, financial mathematics, and RNA-folding). In the recent years, a new characterization has been established for limit laws of random matrices via the spectra of certain random self-adjoint differential operators. This provides a new connection between random matrices and self-adjoint operators more than half a century after Wigner's original idea. This research project explores this new area.This project builds on recent results of the investigator and collaborators on random operator representations of scaling limits of random matrices. The aim is to study the resultant random operators in order to learn more about the limit point processes. Questions under study include the application of the classical theory of differential operators to random matrices, the study of operator level convergence results for random matrix models with quantitative error bounds, and the investigation of the connection between random matrix limits and diffusions in the hyperbolic plane. The project also includes investigations related to the study of the large-scale behavior of interacting stochastic systems. It is conjectured that a wide family of one-dimensional interacting stochastic systems share an unusual scaling behavior with limit distributions related to random matrix theory; this is the Kardar-Parisi-Zhang (KPZ) universality class. The investigator intends to explore various models belonging (or conjectured to belong) to the KPZ universality class, with a special focus on directed polymers, a model describing random paths in a space-time environment that is also random.