Integrable Systems, Integral Operators, and Probabilistic Models

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Natural phenomena are complex. The purpose of mathematical modeling is to study admittedly simpler versions of what one sees in nature and extract exact results which could help explain, and sometimes predict, events in the real world. There are "nonequlibrium processes", in which the system is evolving, and "equlibrium processes" which model the behavior of the system after it has settled down. This project studies one of each, the "Ising model" (an equlibrium process) and the "asymmetric simple exclusion process" (a nonequlibrium process). The first is the simplest (although still quite complicated) description of ferromagnetism. It is the two-dimensional model that is studied, in particular its thermodynamic properties (in which the number of particles gets infinitely large). Magnetism and magnetic materials are pervasive throughout technology, so it is important to achieve exact mathematical results for them. The asymmetric simple exclusion process is the simplest model of phenomena in which particles interact with each other. Precisely, no two can occupy the same site at the same time. The model has far-ranging applications in nonequilibrium statistical physics, engineering, and biological systems. The one-dimensional case is what is studied (think of electrons flowing in a wire). Theoretical results of the principal investigator and collaborator have been confirmed by recent experiment. This project will result in a deeper understanding of this process and related random models in statistical physics.The asymmetric simple exclusion process (ASEP) is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. ASEP is closely related, in a certain scaling limit, to the Kardar-Parisi-Zhang (KPZ) equation, a nonlinear stochastic partial differential equation that is expected to describe a large class of stochastically growing interfaces. Recent work by Sasamoto, Spohn, Amir, Corwin and Quastel, using results of Tracy and the PI, have rigorously established this link between ASEP and the KPZ equation. This project builds on the work of Tracy and the PI and involves: (i) the rigorous operator-theoretic underpinnings of asymptotic results for ASEP with an open boundary; (ii) the study of a discrete time version of ASEP; and (iii) the role of Weyl groups in classifying interacting particle systems solvable by a Bethe Ansatz.