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

研究者研究了两个主题：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。关于无限大维随机方程和统计流体力学，研究者关注三个问题。第一部分关注非线性随机偏微分方程的典型解的结构，如二维随机Navier-Stokes方程和随机burgersequence。第二部分讨论了无界区域上无粘随机强迫方程稳态的唯一性。第三部分是关于随机强迫形式保守系统中能量从高模态向低模态转移的机制。对于这些问题中的每一个，都描述了新的分析方法和新的现象。这项工作的目标是更全面地分析这些现象。在组合学、生长模型和随机矩阵理论中，研究了高斯正交系综（GOE）随机矩阵理论的基本问题，重点研究了对称组合模型与高斯正交系综的极限分布之间的联系。考虑的问题包括物理学中出现的随机平铺问题，随机界面研究中出现的多核生长模型，以及凝聚态物理和电气工程中出现的渗透模型。工作的目标是在GOE随机矩阵理论的背景下证明新的中心极限定理。理解随机强迫流体动力学方程的小尺度结构与湍流理论和统计力学有关。溶液的小尺度结构在理论和实验上都很重要。在空气动力学和流体力学的许多具体问题中出现的湍流中，正是这种结构起着重要作用。这里研究的特殊问题有助于当前对这种小规模结构的理解。随机矩阵理论技术在各种看似不相关的领域中回答了广泛的问题，这一点最近变得很清楚。从本质上讲，随机矩阵理论提供了一个模型来捕捉许多不同过程的波动。在这个项目中研究的问题拓宽了这类问题和过程，并提高了目前对导致随机矩阵统计的各种现象的理解。

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.