Criticality and Nonlinearity in Interacting Particle Systems and Stochastic Partial Differential Equations

The project concerns the time evolution of large, stochastic systems, with a focus on their nonlinear and critical behaviors. The systems studied in this project are representative of natural phenomena, such as crystal growths, evolution of magnetic domains, paths evolving in a noisy environment, and randomly stirred fluids. Often these systems exhibit nonlinearity and criticality. Nonlinearity refers to the behavior that the randomness of the macroscopic observables are related to the underlying microscopic randomness in a nonlinear fashion. Criticality refers to the phenomenon that a given system exhibits drastically distinct macroscopic behaviors when certain parameters in the underlying microscopic model reach some critical values. Time-evolutionary stochastic systems form a large body of probability theory, yet the type of phenomena considered here sits on the frontiers of current standard theories. The goal of this project is to unveil the mathematical structure pertaining to the aforementioned scopes, and to refine the existing theories to study these systems. In concrete terms, the project studies three types of models: interacting particles with moving boundaries, stochastic partial differential equaitons (SPDEs) at their criticality, and stochastic six vertex-types models. Particle systems with moving boundaries give rise to Stefan?s problem in PDE, and in one particular case those systems relate to the critical point of a reaction-diffusion particle system. The principal investigator seeks to develop more robust tools that do not require explicit stationary distributions and apply to the aforementioned critical point. SPDEs exhibit criticality for certain parameters, where the solutions become non-measurable with respect to the driving noise. This research aims at studying the correlation functions, regularity, and local properties of a few specific examples of such SPDEs. The stochastic six-vertex model is a specialization of the ice-type models that can be formulated as a Markov process. It hosts a number of degenerations, including the totally asymmetric simple exclusion process. As a first step toward understanding the limiting shape of these models, the principal investigator plans to study the large deviations utilizing the Markov structures of these models.