Universality and semi-classical behavior in 2+1 dimensional integrable systems and random matrices

Colorado State University, Fort Collins CO

Investigators

Abstract

The understanding and eventual control of complicated phenomena is a primary goal of scientific research. "Universality" refers to robustness of certain phenomena and to the counterintuitive prevalence of the same phenomena across a wide array of different physical situations and models. For example, waves in the ocean can organize themselves into "trains" transporting energy, and analogous trains are also observed in laser beams propagating in optical fibers; statistical fluctuations in nuclear experiments led to a new type of universality that subsequently has been observed in a wide variety of situations modeled with randomness, as far-flung as the statistics of spacings between parked cars! This research project involves the detailed rigorous analysis of canonical models for a wide variety of physical settings whose singular behavior is a guide for the understanding of some complicated phenomena in nature. The project aims to develop methods to understand, predict, and control system behavior. The potential long-term impacts of this research program stem from the emergence of universality as a new paradigm in science: probing its range of applicability is fundamental in emerging areas as well as established ones. This research project concerns the development of new methods for the asymptotic analysis of Riemann-Hilbert problems and d-bar problems and application of these techniques to problems in a range of fields including random matrix theory, nonlinear partial differential equations, orthogonal polynomials, and asymptotic combinatorics. In each of these areas, the overarching goal is to provide a complete description of the system (be it the eigenvalues of a random matrix, or the solution of a nonlinear partial differential equation). Two examples under study are: (1) Eigenvalue statistics in the normal matrix model. The quest for universal behavior when the eigenvalues are accumulating in a two-dimensional region is in its infancy; this project aims to create connections between rigorous mathematical analysis and physical intuition developed over the last 20 years. (2) The asymptotic behavior of the partition function in the two-cut regime, for which ideas and conjectural formulae have existed for some time.