Reflected Brownian motions in the KPZ universality class

Abstract

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, stationary initial conditions, and mixtures thereof. 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. This book serves as an introduction to the KPZ universality class, illustrating the main concepts by means of a single model only. It will be of interest to readers from interacting diffusion processes and non-equilibrium statistical mechanics.

arXiv commentarXiv admin note: text overlap with arXiv:1502.01468

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• Scaling limit for Brownian motions with one-sided collisions
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Cited by in corpus (13)

• Two time distribution in Brownian directed percolation
• Uniform convergence to the Airy line ensemble
• Three-halves variation of geodesics in the directed landscape
• The 1+1 dimensional Kardar-Parisi-Zhang equation: more surprises
• Interlacing Diffusions
• Upper tail bounds for stationary KPZ models
• KPZ fluctuations in finite volume
• Dynamics of finite inhomogeneous particle systems with exclusion interaction
• Reflecting Brownian motion in generalized parabolic domains: explosion and superdiffusivity
• Exact solution of interacting particle systems related to random matrices
• Nonstationary generalized TASEP in KPZ and jamming regimes
• Non-Markovianity of $2K-B$ and a degeneration
• On the duality between particles and polymers