Scaling Limits of Multi-layer Particle Systems

Abstract

This thesis focusses on the study of multi-layer particle systems with an emphasis on the scaling limits of particle systems of this type. Chapter 1 gives an overall introduction to the field of statistical physics and interacting particle systems, and gives a motivation on the study of multi-layer particle systems. In Chapter 2 we introduce the mathematical background required for this thesis in a rigorous manner. The topics discussed include Markov semigroups and generators, path-space convergence, ergodic theory, martingales, couplings, and duality. In Chapter 3 we introduce three types of multi-layer particle systems; the multi-layer exclusion process, the multi-layer inclusion process and the run-and-tumble particle process. We then characterize the ergodic measures with a finite moment condition for these three processes using duality and successful couplings. In Chapter 4 we study the hydrodynamic limit and the stationary fluctuations of the multi-layer run-and-tumble particle process, and use them to infer the same scaling limits for the total density. Furthermore, by an application of Schilder's theorem, we find a large deviation result for the fluctuation field of the total density. In Chapter 5 we establish a large deviation principle for the multi-species stirring process. The method of proof involves studying the hydrodynamic limit of a weakly asymmetric process and a superexponential estimate. In Chapter 6 we return to the multi-layer setting and establish a large deviation principle for the run-and-tumble particle process on two layers with an added mean-field interaction, meaning that the switching between the layers depends on the magnetization of the process. We end with a first step towards an explicit large deviation principle of the total density.