Large Scale Random Systems: Stochastic Partial Differential Equations and Random Matrix Theory

研究人员研究了两个主题：统计流体力学和各种组合模型及其与随机矩阵理论的联系。关于无限维随机方程和统计流体力学，研究者主要关注三个问题。第一部分是关于非线性随机偏微分方程解的结构，如二维随机Navier-Stokes方程和随机Burger序列。第二个问题是关于无界区域上无粘随机强迫方程定态的唯一性。第三部分是随机强迫形式保守系统从高模到低模的能量传递机制。每一个问题都描述了新的分析方法和新的现象。这项工作的一个目标是更全面地分析这些现象。在组合学、增长模型和随机矩阵理论中，研究者研究了高斯正交系综随机矩阵理论的基本问题，重点研究了对称组合模型与限制高斯正交系综分布之间的联系。所考虑的问题包括物理学中出现的随机瓦片问题，随机界面研究中出现的多核增长模型，以及凝聚态物理和电气工程中出现的渗流模型。这项工作的一个目标是在GOE随机矩阵理论的背景下证明新的中心极限定理。了解随机受力水动力方程的小尺度结构与湍流理论和统计力学有关。解决方案的小尺度结构具有理论和实验两方面的重要性。在空气动力学和流体力学的许多具体问题中，这种结构在湍流中起着重要作用。这里研究的具体问题有助于目前对这一小规模结构的理解。最近，随机矩阵理论技术在许多看似不相关的领域中回答了广泛的问题，这一点最近变得清晰起来。从本质上讲，随机矩阵理论提供了一种捕捉许多不同过程的波动的模型。本项目研究的问题包括这类问题和过程，并提高了对导致随机矩阵统计的各种现象的当前理解。

The investigator studies two topics: statisticalhydrodynamics, and various combinatorial models with theirconnections to random matrix theory. With regard to infinitedimensional stochastic equations and statistical hydrodynamics,the investigator focuses on three questions. The first concernsthe structure of typical solutions for nonlinear stochasticpartial differential equations such as the two-dimensionalstochastic Navier-Stokes equations and the stochastic Burgersequation. The second concerns the uniqueness of steady states forinviscid randomly forced equations on unbounded domains. Thethird concerns the mechanism of energy transfer from high to lowmodes in stochastically forced formally conservative systems. Foreach of these questions new analytic approaches and new phenomenahave been described. A goal of this work is to more completelyanalyze these phenomena. In combinatorics, growth models, andrandom matrix theory the investigator studies basic questions ofGaussian Orthogonal Ensemble (GOE) random matrix theory, focusingon the connection of symmetrized combinatorial models to limitingdistributions of the Gaussian orthogonal ensemble. The questionsconsidered include problems of random tiling that arise inphysics, polynuclear growth models that arise in the study ofrandom interfaces, and percolation models that arise in condensedmatter physics and electrical engineering. A goal of the work isto prove new central limit theorems in the context of GOE randommatrix theory. Understanding the small-scale structure of stochasticallyforced hydrodynamic equations is related to turbulence theory andstatistical mechanics. The small-scale structure of solutions isof both theoretical and experimental importance. It is thisstructure that matters in turbulent flows arising in many concreteproblems of aerodynamics and fluid mechanics. The particularquestiuons studied here contribute to the current understanding ofthis small-scale structure. It has recently become clear thatrandom matrix theory techniques answer a wide range of questionsin a variety of seemingly unrelated fields. Essentially, randommatrix theory affords a model that captures the fluctuations ofmany distinct processes. The questions studied in this projectwiden this class of problems and processes and improve the currentunderstanding of the various phenomena that lead to random matrixstatistics.