CAREER: Random eigenvalue problems and fluctuations of large stochastic systems

PI将研究与随机矩阵理论和大型随机系统波动理论相关的问题。随机矩阵理论的基本问题之一是分析大型随机矩阵的特征值。PI将专注于与β集合的渐近谱特性相关的问题，β集合是一系列模型，概括了一些研究最多的经典随机矩阵模型（例如GOE，GUE，GSE，Wishart）。该项目建立在PI和合作者的最新结果的基础上，其中一般β系综的点过程标度限制是在大部分频谱中推导出来的。所提出的问题包括方法的扩展到其他随机矩阵集成，分析极限点过程和寻找连接到现有的经典情况下的描述。该方法依赖于随机三对角矩阵的研究，这也提供了一个自然的框架，调查某些随机薛定谔算子的渐近谱性质。非严格的物理标度参数表明，一个大类的一维随机系统具有不寻常的超扩散波动的标度指数为2/3。例子包括相互作用的粒子系统，生长模型和定向聚合物。这些系统作为自然科学和社会科学中各种现象的模型而出现。最近十年带来了数学方面的突破：对于某些模型，波动指数已被严格确定，在某些情况下，甚至标度极限已被证明。PI将努力将严格的结果扩展到更广泛的模型家族。特别是某些格子气体的涨落理论，定向聚合物模型和自避免过程将被研究。在建议中考虑的问题处理与复杂的相互作用的大量随机组件的系统的分析。我们可以理解大型随机矩阵元素的联合分布，但特征值的分布通常要复杂得多。相互作用粒子系统的演化在局部水平上可能是简单的，我们可以理解在固定时间内波动的行为，但波动场的标度极限可能是一个非常重要的对象。该提案的目标是开发新的工具，并更好地了解这些问题。该提案的教育部分侧重于加强PI主办机构的本科生和研究生概率教育，并积极让研究生和博士后研究员参与PI的研究领域。

The PI will work on problems related to random matrix theory and the fluctuation theory of large stochastic systems. One of the basic problems of random matrix theory is to analyze the eigenvalues of large random matrices. The PI will focus on problems related to the asymptotic spectral properties of the beta ensembles, a family of models generalizing some of the most studied classical random matrix models (e.g. GOE, GUE, GSE, Wishart). The project builds on the recent results of the PI and collaborators in which the point process scaling limits of general beta ensembles are derived in the bulk of the spectrum. The proposed problems include the extension of methods to other random matrix ensembles, analyzing the limiting point processes and finding connections to the existing descriptions of classical cases. The approach relies on the study of random tridiagonal matrices which also provides a natural framework to investigate the asymptotic spectral properties of certain random Schrodinger operators. Non-rigorous physical scaling arguments suggest that a large class of one-dimensional random systems have unusual superdiffusive fluctuations with a scaling exponent of 2/3. Examples include interacting particle systems, growth models and directed polymers. These systems arise as models for various phenomena in the natural and social sciences. The recent decade brought a breakthrough from the mathematical side: for certain models the fluctuation exponent has been determined rigorously and in some cases even scaling limits have been proved. The PI will work towards extending the rigorous results to a wider family of models. In particular the fluctuation theory of certain lattice gases, directed polymer models and self avoiding processes will be studied.The problems considered in the proposal deal with the analysis of systems with a large number of random components with complex interactions. We might understand the joint distribution of the entries of a large random matrix, but the distribution of the eigenvalues is usually a lot more complicated. The evolution of an interacting particle system might be simple on a local level, and we might understand the behavior of the fluctuations for a fixed time, but the scaling limit of the fluctuation field may be a highly non-trivial object. The goal of the proposal is to develop new tools and to provide a better understanding for such problems. The educational part of the proposal focuses on enhancing undergraduate and graduate probability education at the PI's host institution and actively involving graduate students and postdoctoral fellows in the PI's area of research.