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

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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.