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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