Random Matrices and Interacting Systems

在20世纪50年代，物理学家尤金维格纳（Eugene Wigner）提出研究大型随机对称矩阵，以近似复杂自伴算子的行为。他的目标是了解随机矩阵的本征值的标度特性，以了解某些运营商的频谱所产生的研究重核。在最近的半个世纪中，随机矩阵在数学中的广泛领域（例如，组合学、数论和相互作用随机系统的研究）以及各种其它领域（例如，信息论、金融数学和RNA折叠）。近年来，利用一类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。这提供了一个新的连接随机矩阵和自伴运营商超过半个世纪后维格纳的原始想法。本研究计画探讨这一新领域，并建立在研究者与合作者最近关于随机矩阵标度极限的随机算子表示的研究成果上。目的是研究所得的随机算子，以便更多地了解极限点过程。 正在研究的问题包括应用经典理论的微分算子随机矩阵，研究运营商水平的收敛结果随机矩阵模型与定量误差界，并调查随机矩阵的限制和扩散之间的连接在双曲平面。该项目还包括与相互作用的随机系统的大规模行为的研究相关的调查。它是一个广泛的家庭一维相互作用的随机系统共享一个不寻常的标度行为与随机矩阵理论的极限分布，这是Kardar-Parisi-Zhang（KPZ）普适类。 研究人员打算探索属于（或被证明属于）KPZ普适性类的各种模型，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.