The investigator studies two topics: statistical hydrodynamics, and various combinatorial models with their connections to random matrix theory. With regard to infinite dimensional stochastic equations and statistical hydrodynamics, the investigator focuses on three questions. The first concerns the structure of typical solutions for nonlinear stochastic partial differential equations such as the two-dimensional stochastic Navier-Stokes equations and the stochastic Burgers equation. The second concerns the uniqueness of steady states for inviscid randomly forced equations on unbounded domains. The third concerns the mechanism of energy transfer from high to low modes in stochastically forced formally conservative systems. For each of these questions new analytic approaches and new phenomena have been described. A goal of this work is to more completely analyze these phenomena. In combinatorics, growth models, and random matrix theory the investigator studies basic questions of Gaussian Orthogonal Ensemble (GOE) random matrix theory, focusing on the connection of symmetrized combinatorial models to limiting distributions of the Gaussian orthogonal ensemble. The questions considered include problems of random tiling that arise in physics, polynuclear growth models that arise in the study of random interfaces, and percolation models that arise in condensed matter physics and electrical engineering. A goal of the work is to prove new central limit theorems in the context of GOE random matrix theory.
Understanding the small-scale structure of stochastically forced hydrodynamic equations is related to turbulence theory and statistical mechanics. The small-scale structure of solutions is of both theoretical and experimental importance. It is this structure that matters in turbulent flows arising in many concrete problems of aerodynamics and fluid mechanics. The particular questiuons studied here contribute to the current understanding of this small-scale structure. It has recently become clear that random matrix theory techniques answer a wide range of questions in a variety of seemingly unrelated fields. Essentially, random matrix theory affords a model that captures the fluctuations of many distinct processes. The questions studied in this project widen this class of problems and processes and improve the current understanding of the various phenomena that lead to random matrix statistics.
