The proposed research project is focused on the study of random matrices and random walks evolving in random media. Much of the existing theory of random matrices deals with normal matrices, which are stable under perturbations; many of the existing tools implicitly use either this stability, or some strong independence assumptions, even in deriving rough results. A major goal of the present proposal is to build on previous work of the PI and collaborators and develop techniques that work even in a context where both structural assumptions (normality or independence) fail. While the proposal discusses and addresses some very specific questions, it is part of a larger attempt to develop new techniques that would be applicable to a wide range of RMT questions. The second topic to be studied concerns random walks and branching processes in random environments. Though the theory of random walks is well developed, many gaps in understanding remain when one changes the medium in which the walk evolves to a random medium. In spite of rapid progress that was achieved in the last few years by several researchers, many fundamental questions remain unanswered. The proposed project will built on previous work, by the PI and other researchers, with the goal of resolving some of these outstanding questions.
Matrices are the cornerstone of linear algebra, and are fundamental building blocks in operator theory. Random matrix theory (RMT) is concerned with the study of properties of random matrices, typically in the limit where the dimension of the matrix is large. Motivations and applications for this study come from several areas of mathematics (probability theory, number theory, representation theory), the physical sciences (especially, mathematical physics), statistics, and engineering (specifically, communication and information theories). In recent years, RMT has emerged as a major research area within mathematics, combining techniques from probability theory, operator algebras, complex analysis, and combinatorics. Several major (and fundamental) open questions have recently been resolved, especially concerning the universality of limit laws regarding the spectrum of large random matrices. The proposed research will seek to significantly expand the theory toward a class of matrices whose spectrum is not stable under (small) perturbations. While the focus of the research is theoretical, an impact on applications is expected, e.g. in evaluating the stability of complex systems, in computations related to quantum information theory, and in statistics. The second main focus of the study, random walks, are arguably the stochastic processes most studied by mathematicians, having the widest range of applications in fields as diverse as the physical sciences, engineering, and the social sciences. Though the theory of random walks is by now well developed, this is not at all the case when one changes the medium in which the walk evolves to a random medium, thus obtaining a random walk in random environment (RWRE). Such RWRE's can be used to model many problems of motion in random media in the physical and engineering sciences, and are mathematically appealing because on the one hand the model is very simply stated, while on the other hand established tools for studying processes in random media are not applicable in the study of RWRE. The goal of the current proposal is to develop new basic probabilistic techniques that will allow to make provable predictions concerning the behavior of RWRE. It is expected that such techniques will be useful in the study of other processes, and link naturally to the study of trapping models and reinforced random walks. While not explicitly targeted in this proposal, trapping models have recently been used to study the environmental impact of nuclear waste, and RWRE's can naturally be used in this context to model the spread of contamination in a real environment, once the required mathematical background and tools are in place.
