Reflected Brownian Motions in the KPZ Universality Class

Herbert Spohn, Patrik Ferrari, Thomas Weiss, et al.

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Naturwissenschaften, Medizin, Informatik, Technik / Sonstiges

Beschreibung

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