Random Matrix Models and Statistical Mechanics

Indiana University, Bloomington IN

Investigators

Abstract

This research project has an interdisciplinary character on the frontier between physics and mathematics. The project investigates the phenomena of scaling and universality in random matrices, which are central in many areas of modern science. Improved understanding of these phenomena will have a significant impact on both mathematical theory and its applications. The project aims to develop new and powerful mathematical approaches involving methods from different areas of mathematics including analysis, the theory of integrable systems, probability theory, semiclassical asymptotics, and complex analysis. The results of the project will have direct applications to combinatorial aspects of quantum gravity, exactly solvable models of statistical mechanics, spin systems on random surfaces, theory of critical phenomena and phase transitions, and quantum chaos. Further applications include the theory of knots and links and related problems in molecular biology, growth models, and statistical data analysis. This research project is directed at fundamental problems of the theory of random matrix models and statistical mechanics. The project investigates different conjectures of universality of the scaling limits of eigenvalue correlation functions and exactly solvable models of statistical physics and hydrodynamics. This includes: (a) the development of the Riemann-Hilbert approach to random normal matrix models with applications to Hele-Shaw flows in hydrodynamics, Laplacian growth, and quantum Hall effect; (b) the development of the Riemann-Hilbert approach to random matrix models with external source, including the problems of phase transitions, critical phenomena, and double scaling limits in these models; (c) the exact solution of the six-vertex model with various boundary conditions in different phases; and (d) the study of the phase separation in the six-vertex model.