Reflected Brownian Motions in the KPZ Universality Class

  • Thomas Weiss
  • Patrik Ferrari
  • Herbert Spohn

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

Table of contents

  1. Front Matter
    Pages i-vii
  2. Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 1-7
  3. Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 9-23
  4. Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 25-30
  5. Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 31-43
  6. Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 45-70
  7. Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 71-95
  8. Thomas Weiss, Patrik Ferrari, Herbert Spohn
    Pages 97-118

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


Kardar-Parisi-Zhang equation stochastic PDE Bethe Ansatz Skorokhod construction last passage percolation determinantal point process Fredholm determinants Eynard-Metha Theorem determinantal structure Poisson initial conditions Airy proceses steepest descent contour integration polynuclear growth model TASEP model totally asymmetric simple exclusion process quantum integrability

Authors and affiliations

  • Thomas Weiss
    • 1
  • Patrik Ferrari
    • 2
  • Herbert Spohn
    • 3
  1. 1.Zentrum MathematikTechnische Universität MünchenGarchingGermany
  2. 2.Institut für Angewandte MathematikUniversität BonnBonnGermany
  3. 3.Zentrum Mathematik, M5Technische Universität MünchenMunichGermany

Bibliographic information

  • DOI
  • Copyright Information The Author(s) 2017
  • Publisher Name Springer, Cham
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-3-319-49498-2
  • Online ISBN 978-3-319-49499-9
  • Series Print ISSN 2197-1757
  • Series Online ISSN 2197-1765
  • Buy this book on publisher's site
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