The main activity of the project is the analysis of limit laws for certain stochastic growth models/interacting particle systems and magnetic spin chains: the asymmetric simple exclusion process (ASEP) and the quantum-mechanical Heisenberg-Ising spin chain. The objective is to use the techniques of integrable systems, combinatorics, operator theory, and asymptotic analysis to understand, and to derive new results for, these probabilkistic models. One of the basic problems of ASEP is the study of height (or current) fluctuations, in particular to classify various limiting distributions that arise. It is conjectured (KPZ universality) that the fluctuations are universal for a large class of growth models. The PI and Craig A. Tracy established KPZ universality for ASEP with step initial condition, a result that required some new combinatorial identities. The project proposes the study of other initial conditions, in particular Bernoulli initial condition, which would require a better understanding of the combinatorial identities. For the Heisenberg-Ising model the focus will be on the dynamics of domain wall growth. The method will entail a novel use of the Bethe Ansatz that avoids the awkward spectral theory of the Heisenberg-Ising Hamiltonian.
The project would have broad impact in other areas of mathematics and science.The ideas and techniques coming from random matrix theory and integrable systems have had an impact on such diverse subjects as probability, statistics, biostatistics, number theory , condensed matter physics, and engineering. In particular, the ASEP model has applications in nonequilibrium statistical physics and biological systems. In fact, the T(otally)ASEP was introduced in the late 1960s as a model of ribosome motion along mRNA. The understanding of the limit laws that arise, in particular the underlying reason for their ubiquitous occurrence, will have great value in these areas. These universal distributions were first discovered and computed in the mathematical literature and have since found many applications. It is fully expected that elucidation of these laws in ASEP will further impact the applied areas -- not only to create new techniques to solve long-standing problems in the above-mentioned fields but also to proviude new and unexpected applications.
