Random Matrices and Interacting Systems

在20世纪50年代，物理学家尤金·维格纳（Eugene Wigner）提出了对大型随机对称矩阵的研究，以近似复杂自伴随算子的行为。他的目标是了解随机矩阵的特征值的标度特性，以深入了解从重核研究中产生的某些算子的谱。在过去的半个世纪里，随机矩阵在数学（如组合学、数论和相互作用随机系统的研究）和其他领域（如信息论、金融数学和rna折叠）中得到了广泛的应用。近年来，利用随机自伴随微分算子的谱，建立了随机矩阵极限律的一个新的表征。这为随机矩阵和自伴随算子之间提供了一种新的联系，比Wigner最初的想法晚了半个多世纪。这个研究项目探索了这个新领域。这个项目建立在研究者和合作者最近关于随机矩阵缩放极限的随机算子表示的结果之上。目的是研究由此产生的随机算子，以便更多地了解极限点过程。研究的问题包括经典微分算子理论在随机矩阵中的应用，具有定量误差界的随机矩阵模型的算子级收敛结果的研究，以及双曲平面上随机矩阵极限与扩散之间的联系的研究。该项目还包括与相互作用随机系统的大规模行为研究相关的调查。推测了一类一维相互作用随机系统具有与随机矩阵理论相关的极限分布的不寻常的标度行为；这是kardar - paris - zhang （KPZ）通用性类。研究者打算探索属于（或推测属于）KPZ普适类的各种模型，特别关注定向聚合物，一种描述时空环境中随机路径的模型，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.