Reflected Brownian motions in the KPZ universality class

TL;DR

This work provides a rigorous analysis of a one-dimensional interacting Brownian motion system within the KPZ universality class, demonstrating universal properties and convergence to Airy processes under various initial conditions.

Contribution

It introduces a detailed, rigorous study of reflected Brownian motions in the KPZ class, highlighting universal behaviors and convergence results for different initial conditions.

Findings

Convergence of scaled processes to Airy processes

Universal properties depend on initial conditions

Rigorous analysis of singular interactions in Brownian motions

Abstract

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, stationary initial conditions, and mixtures thereof. 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. This book serves as an introduction to the KPZ universality class, illustrating the main concepts by means of a single model only. It will be of interest to readers from interacting diffusion processes and non-equilibrium statistical mechanics.

Full text

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**Reflected Brownian motions

in the KPZ universality class

**

Thomas Weiss

Patrik Ferrari

Herbert Spohn

September 2016

Chapter 1 Abstract

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 LABEL:secPP), stationary initial conditions (Chapter LABEL:secPoi), and mixtures thereof (Chapter LABEL:secMixed). 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. This book serves as an introduction to the KPZ universality class, illustrating the main concepts by means of a single model only. It will be of interest to readers from interacting diffusion processes and non-equilibrium statistical mechanics.

Contents

1. 1 Abstract