Reflected Brownian Motions in the KPZ Universality Class

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Springer, Dec 16, 2016 - Science - 118 pages
This book presents a detailed study of a system of interacting Brownian motions in one dimension. The interaction is point-like such that the n-th Brownian motion is reflected from the Brownian motion with label n-1. This model belongs to the Kardar-Parisi-Zhang (KPZ) universality class. In fact, because of the singular interaction, many universal properties can be established with rigor. They depend on the choice of initial conditions. Discussion addresses packed and periodic initial conditions (Chapter 5), stationary initial conditions (Chapter 6), and mixtures thereof (Chapter 7). The suitably scaled spatial process will be proven to converge to an Airy process in the long time limit. A chapter on determinantal random fields and another one on Airy processes are added to have the notes self-contained. These notes serve as an introduction to the KPZ universality class, illustrating the main concepts by means of a single model only. The notes will be of interest to readers from interacting diffusion processes and non-equilibrium statistical mechanics.
 

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Contents

1 Introduction
1
2 OneSided Reflected Brownian Motions and Related Models
9
3 Determinantal Point Processes
25
4 Airy Processes
31
5 Packed and Periodic Initial Conditions
45
6 Stationary Initial Conditions
71
7 More General Initial Conditions and Their Asymptotics
97
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