The PI will work on problems related to random matrix theory and the fluctuation theory of large stochastic systems. One of the basic problems of random matrix theory is to analyze the eigenvalues of large random matrices. The PI will focus on problems related to the asymptotic spectral properties of the beta ensembles, a family of models generalizing some of the most studied classical random matrix models (e.g. GOE, GUE, GSE, Wishart). The project builds on the recent results of the PI and collaborators in which the point process scaling limits of general beta ensembles are derived in the bulk of the spectrum. The proposed problems include the extension of methods to other random matrix ensembles, analyzing the limiting point processes and finding connections to the existing descriptions of classical cases. The approach relies on the study of random tridiagonal matrices which also provides a natural framework to investigate the asymptotic spectral properties of certain random Schrodinger operators. Non-rigorous physical scaling arguments suggest that a large class of one-dimensional random systems have unusual superdiffusive fluctuations with a scaling exponent of 2/3. Examples include interacting particle systems, growth models and directed polymers. These systems arise as models for various phenomena in the natural and social sciences. The recent decade brought a breakthrough from the mathematical side: for certain models the fluctuation exponent has been determined rigorously and in some cases even scaling limits have been proved. The PI will work towards extending the rigorous results to a wider family of models. In particular the fluctuation theory of certain lattice gases, directed polymer models and self avoiding processes will be studied.
The problems considered in the proposal deal with the analysis of systems with a large number of random components with complex interactions. We might understand the joint distribution of the entries of a large random matrix, but the distribution of the eigenvalues is usually a lot more complicated. The evolution of an interacting particle system might be simple on a local level, and we might understand the behavior of the fluctuations for a fixed time, but the scaling limit of the fluctuation field may be a highly non-trivial object. The goal of the proposal is to develop new tools and to provide a better understanding for such problems. The educational part of the proposal focuses on enhancing undergraduate and graduate probability education at the PI's host institution and actively involving graduate students and postdoctoral fellows in the PI's area of research.
