Integrable Systems, Operator Determinants, and Probabilistic Models

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The focus of this project is the analysis of limit laws for certain stochastic growth models. The main growth model considered is the interacting particle system called the asymmetric simple exclusion process. This model is a widely studied model in probability theory and theoretical physics since it is one of the simplest models of nonequilibrium behavior that exhibits nonclassical fluctuations. In this project the asymmetric simple exclusion process will be studied for a variety of different initial conditions. The methods use ideas from Bethe Ansatz, Yang-Baxter equations as well as techniques from operator theory and combinatorics. Generalizations of the model to more than one species will also be part of the project. It is expected that these limit laws will have a "universal" behavior; and will, in fact, describe the fluctuations in a much larger class of growth models.This project involves the study of current fluctuations in the asymmetric simple exclusion process for a variety of initial configurations. This is a model of interacting particles on a one-dimensional lattice. The model has attracted wide attention from both mathematicians and physicists since it is one of the simplest models to incorporate far from equilibrium behavior with nonclassical fluctuations. These fluctuations are expected to have a new universal behavior similar in their applicability to the famous bell-shaped curve (the Gaussian distribution) of classical probability. A long-term goal of research in this area is the establishment of new limit laws similar in nature to the classical central limit theorem. Already these new universal distributions are being applied to various problems in growth processes, population genetics, and finance. This project will extend our knowledge of fluctuations to a much wider class of growth models.