## Reflected Brownian Motions in the KPZ Universality ClassThis 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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### Contents

1 | |

2 OneSided Reflected Brownian Motions and Related Models | 9 |

3 Determinantal Point Processes | 25 |

4 Airy Processes | 31 |

5 Packed and Periodic Initial Conditions | 45 |

6 Stationary Initial Conditions | 71 |

7 More General Initial Conditions and Their Asymptotics | 97 |

### Other editions - View all

Reflected Brownian Motions in the KPZ Universality Class Thomas Weiss,Patrik Ferrari,Herbert Spohn No preview available - 2017 |

Reflected Brownian Motions in the KPZ Universality Class Thomas Weiss,Patrik Ferrari,Herbert Spohn No preview available - 2016 |

### Common terms and phrases

Airy function Airy processes Airy2 Airystal Airystat asymptotic analysis Baik Bethe ansatz Borodin bounded choose const contour convergence correlation Corwin define the rescaled Definition density Dyson eigenvalue exponential finite finite-dimensional distributions fixed fluctuations Fredholm determinant function Gaussian given initial condition x(0 integral integrand Johansson joint distributions kernel KPZ equation KPZ universality class Lambert W function last passage percolation Math matrix motions with initial numbers one-sided reflected Brownian P.L. Ferrari packed initial conditions parameter particles Phys pointwise convergence Poisson Poisson process Prähofer Probab Proof of Proposition proof of Theorem Proof Proof Proposition 6.2 Quastel random matrices random variables reflected Brownian motions rescaled process ſ dx Sasamoto scaling limit SpringerBriefs in Mathematical Stat stationary steep descent superexponentially system of one-sided TASEP Theorem 6.1 trace class Tracy-Widom distribution zero