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Reflected Brownian Motions in the KPZ Universality Class

Part of the SpringerBriefs in Mathematical Physics Book Series (volume 18)

DOI: 10.1007/978-3-319-49499-9

ISBN: 978-3-319-49498-2

Publisher: Springer International Publishing - Springer Journals

Dec 19, 2016

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Chapter 1Introduction

Weiss, Thomas; Ferrari, Patrik; Spohn, Herbert

Pages: 1 –7

[Back in 1931 Hans Bethe diagonalized the hamiltonian of the one-dimensional Heisenberg spin chain through what is now called the “Bethe ansatz” Bethe (1931).]
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Chapter 2One-Sided Reflected Brownian Motions and Related Models

Weiss, Thomas; Ferrari, Patrik; Spohn, Herbert

Pages: 9 –23

[For reflected Brownian motions with index set Z+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt}...
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Chapter 3Determinantal Point Processes

Weiss, Thomas; Ferrari, Patrik; Spohn, Herbert

Pages: 25 –30

[Determinantal point processes are the main tool for the study of reflected Brownian motions. Thereby marginal distributions can be expressed in terms of Fredholm determinants, a form which is well suited for an asymptotic analysis. However, only partial aspects of the underlying theory of...
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Chapter 4Airy Processes

Weiss, Thomas; Ferrari, Patrik; Spohn, Herbert

Pages: 31 –43

[Airy processes arise through a scaling limit of our system of reflected Brownian motions. Their detailed structure depends on the initial conditions. For the various Airy processes appearing in the text we list here the definitions and, in particular, discuss their interrelation.]
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Chapter 5Packed and Periodic Initial Conditions

Weiss, Thomas; Ferrari, Patrik; Spohn, Herbert

Pages: 45 –70

[Packed initial conditions turn out to be the most readily accessible case. Although such initial conditions have been studied extensively in the literature, we provide here a complete proof, introducing essentially all methods required later on in more complicated situations. The second...
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Chapter 6Stationary Initial Conditions

Weiss, Thomas; Ferrari, Patrik; Spohn, Herbert

Pages: 71 –95

[As stated more precisely in Proposition 2.7, stationary initial conditions are given through a Poisson point process with uniform density. Our tools can be adjusted to also cover such random initial data. The novel difficulty comes from the determinantal expressions being seemingly ill-defined....
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Chapter 7More General Initial Conditions and Their Asymptotics

Weiss, Thomas; Ferrari, Patrik; Spohn, Herbert

Pages: 97 –118

[Besides the three basic initial conditions analyzed in the previous two chapters, a further class of natural initial measures is obtained by first restricting the basic measures to a half-space and then joining pairwise. They all satisfy an asymptotic theorem, where the limit process is given by...

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