Universality in random matrices and integrable systems: asymptotic analysis via Riemann-Hilbert and d-bar methods

University Of Arizona, Tucson AZ

Investigators

Abstract

McLaughlin's research concerns the development of new methods for the asymptotic analysis of Riemann-Hilbert problems and d-bar problems, and to apply the full body of such techniques to cutting edge (as well as classical) problems in a range of fields including random matrix theory, nonlinear partial differential equations, orthogonal polynomials, and asymptotic combinatorics. One driving feature is the exploration of universal behavior in these areas. McLaughlin's previous work in nonlinear partial differential equations has concerned the study of violent oscillations in somewhat specialized ``integrable'' nonlinear partial differential equations. Very recently, conjectures have emerged, analogous to the famous universality conjectures in random matrix theory, which posit that the microscopic generation of these oscillations is universal over entire families of nonlinear partial differential equations. In random matrix theory, universality refers to the phenomenon that eigenvalue statistics of random matrices tend, as the size of the matrices grow to infinity, to behave in a universal way, independent of the exact form of the probabilistic laws used to generate the random matrices. McLaughlin proposes to extend the current realm of universality substantially, in nonlinear partial differential equations and random matrix theory, and to probe the limits of universality: what conditions on V are sufficient to see a deviation from the typical universal behavior? A variety of research projects are also proposed whose aim is the training of graduate students. The understanding and eventual control of complicated phenomena is a primary goal of scientific research. Through this fundamental goal we enhance our ability for technological advancement. Physical models for complex nonlinear phenomena often boil down to the study of partial differential equations in parameter regimes where their solutions exhibit singularly wild behavior. In other instances, statistical theories with great amount of randomness are developed to understand complex phenomena. ``Universality'' refers to robustness of certain phenomena, and to the counter-intuitive prevalence of the same phenomena across a wide array of different physical situations and models. Two examples: (1) Waves in the ocean can organize themselves into "trains" transporting energy, and analogous trains are also observed in laser beams propagating in optical fibers. (2) Statistical fluctuations of nuclear resonance levels measured in the 1950s led to a new type of universality, and the same statistical behavior has been observed in a wide variety of situations modeled with randomness, as far-flung as the statistics of spacings between parked cars! Scientists' ability to predict dramatic behavior through the analysis of such general nonlinear partial differential equations, or statistical theories, is limited. However, there is a class of canonical models for a wide variety of physical settings. Their singular behavior is a guide for the understanding of some complicated phenomena in nature. Some of these are partial differential equations, others are statistical models, but the unifying feature of these models is that researchers are making great progress in their analysis. McLaughlin's research involves the detailed rigorous analysis of these models; he (with collaborators) is developing methods to understand, predict, and control their behavior. The broader impacts of this research program stem from the emergence of universality as a new paradigm in nonlinear science: probing its range of applicability is fundamental in emerging areas as well as established ones. Another very important aspect of the proposed research is the continued training of graduate students in these rapidly developing areas.