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Reflected Brownian Motions in the KPZ Universality Class

  • Book
  • © 2017

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Authors:
  1. Thomas Weiss

    1. Zentrum Mathematik, Technische Universität München, Garching, Germany

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  2. Patrik Ferrari

    1. Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany

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    You can also search for this author in PubMed   Google Scholar

  3. Herbert Spohn

    1. Zentrum Mathematik, M5, Technische Universität München, Munich, Germany

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Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 18)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-vii
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  2. Introduction

    • Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 1-7

  3. One-Sided Reflected Brownian Motions and Related Models

    • Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 9-23

  4. Determinantal Point Processes

    • Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 25-30

  5. Airy Processes

    • Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 31-43

  6. Packed and Periodic Initial Conditions

    • Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 45-70

  7. Stationary Initial Conditions

    • Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 71-95

  8. More General Initial Conditions and Their Asymptotics

    • Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 97-118
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Keywords

About this book

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 modelonly. The notes will be of interest to readers from interacting diffusion processes and non-equilibrium statistical mechanics.

Authors and Affiliations

  • Zentrum Mathematik, Technische Universität München, Garching, Germany

    Thomas Weiss

  • Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany

    Patrik Ferrari

  • Zentrum Mathematik, M5, Technische Universität München, Munich, Germany

    Herbert Spohn

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