Random Matrices and Interacting Systems

随机矩阵和交互系统

基本信息

批准号：
1712551
负责人：
Benedek Valko
金额：
$ 27万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712551&HistoricalAwards=false
关键词：
Random Matrices Interacting Systems

项目摘要

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.

在1950年代，物理学家Eugene Wigner提出了对大型随机对称矩阵的研究，以近似复杂的自我接合操作员的行为。他的目标是了解随机矩阵的特征值的缩放特性，以深入了解重核研究引起的某些操作员的光谱。在过去的半个世纪中，随机矩阵在数学（例如组合学，数字理论和相互作用随机系统的研究）和其他各个领域（例如信息理论，金融数学和RNA折叠）中发现了广泛的领域应用。近年来，已经通过某些随机自我接合差异操作员的光谱为随机矩阵的极限定律建立了新的特征。在Wigner的最初想法之后，这提供了随机矩阵与自动伴侣操作员之间的新联系。该研究项目探讨了这个新领域。该项目基于研究者和合作者的最新结果，即随机矩阵的缩放限制限制的随机操作员表示。目的是研究所得的随机操作员，以了解有关极限点过程的更多信息。研究中的问题包括将差分运算符的经典理论应用于随机矩阵，对具有定量误差界定的随机矩阵模型的操作员级收敛结果的研究，以及对双曲机平面中随机矩阵限制和扩散之间的连接的研究。该项目还包括与研究相互作用随机系统的大规模行为有关的研究。据推测，一个一维相互作用随机系统的广泛家族具有与随机矩阵理论相关的极限分布的异常缩放行为。这是Kardar-Parisi-Zhang（KPZ）通用类。研究人员打算探索属于KPZ通用类别的各种模型（或猜想），特别关注定向聚合物，该模型在时空环境中描述随机路径的模型也是随机的。