Shock Fluctuations in KPZ Growth Models

Abstract

The Kardar-Parisi-Zhang (KPZ) universality class is a class of stochastic growth models which has attracted much interest, especially since the discovery about 15 years ago that the Tracy-Widom distributions from random matrix theory arise in it. Since then, more and more subclasses of the KPZ class have been studied, and experimental evidence for the soundness of KPZ scalings and statistics has been given. <br /> The aims of this thesis are the following. First, we introduce the KPZ class and discuss its conjectured universal scaling properties, limiting distributions and processes.As examples of growth models belonging to the KPZ class where these aspects have been studied, we treat in particular the (totally) asymmetric simple exclusion process ((T)ASEP) and last passage percolation (LPP). We describe the Tracy-Widom distributions, and the Airy processes which appear in these models. As a first result, we obtain the limiting distribution of certain particle positions in TASEP with particular initial data. <br /> Second, we focus on the study of shocks. After introducing the main concepts, we prove the emergence of an independence structure, which appears on a general level in LPP. With this independence, we provide the limiting distributions of shock positions in concrete cases in TASEP and show that they are given by products of Tracy-Widom distributions. We also show that the correlation length in KPZ models, which in all settings considered so far was t to the power 2/3 (t being the observation time), degenerates at the shock to t to the power 1/3.<br /> Finally, we consider a critical scaling, which, depending on the choice of the parameter, interpolates between shocks, flat profiles, and rarefaction fans. We prove that the fluctuations of particle positions in this critical scaling are, in the large time limit, given by a new transition process. The correlation length is shown to be t to the power 2/3 again. We perform a numerical study which suggests that we recover the product structure of shocks by letting the scaling parameter tend to infinity.