he asymmetric simple exclusion process (ASEP) is a model of interacting particles on a lattice. ASEP is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. As such the model has attracted much attention from both mathematicians and physicists. This award will support (1) the exploration of the underlying integrable structure of ASEP using spectral methods, (2) the generalization of ASEP to multi-species where there are now second-class, third-class, etc. particles, and (3) ASEP on the half-line. The integrable structure referred to commonly goes under the name ``Bethe Ansatz''; and in (3) this project will support the analysis of the new structures required in restricting to a half-line. (Here the familiar sum over permutations in Bethe Ansatz is replaced by a sum over signed permutations.) A closely related model is the quantum spin model called the Heisenberg-Ising model. This project will support the analysis of the domain wall problem in the Heisenberg-Ising chain. A long-term goal of this project is the study of limit laws and their universality in these various models.
The Gaussian distribution (the familiar bell-shaped curve) is important because of its universality; that is, it applies to a wide variety of seemingly unrelated problems. The underlying common theme for all these problems is the fact that the objects under study have some degree of ``independence''; or stated in more physical terms, are non-interacting (or weakly interacting). These types of problems are well-understood both physically and mathematically. Current research involves processes which are strongly interacting; and as such, there is no satisfactory general theory as there exists for the non-interacting cases. In the search for new universal laws, the ideas, methods and results of random matrix theory, interacting particle systems and their associated limit theorems; in particular, the Tracy-Widom (TW) distributions, have found an ever widening impact in science and engineering. Since their initial discovery in random matrix theory, the TW distributions have been shown to describe the properties of a number of random systems including the fluctuations of a large class of growing interfaces and directed polymer systems. These same TW distributions are now regularly used in multivariate statistical analysis where they have become a standard statistical tool for inferring population structure from genetic data. Their use in engineering involving signal analysis and wireless communications is just beginning. This project extends the scope of these theoretical investigations; and as such, will make available to a wider audience the resulting mathematical results.
