Weakly asymmetric bridges and the KPZ equation

TL;DR

This paper studies the weakly asymmetric corner growth model on discrete bridges, revealing the hydrodynamic limit as the inviscid Burgers equation and identifying KPZ fluctuations for certain asymmetry regimes.

Contribution

It provides a complete asymptotic analysis of the model, including the transition of KPZ fluctuations and their sudden disappearance at a critical asymmetry parameter.

Findings

Hydrodynamic limit given by inviscid Burgers equation with zero-flux boundary conditions.

KPZ fluctuations occur for asymmetry parameter  in (0,1/3].

KPZ fluctuations vanish at a deterministic time when =1/3.

Abstract

We consider the corner growth dynamics on discrete bridges from $(0,0)$ to $(2N,0)$, or equivalently, the weakly asymmetric simple exclusion process with $N$ particles on $2N$ sites. We take an asymmetry of order $N^{-\alpha}$ with $\alpha \in (0,1)$ and provide a complete description of the asymptotic behaviour of this model. In particular, we show that the hydrodynamic limit of the density of particles is given by the inviscid Burgers equation with zero-flux boundary condition. When the interface starts from the flat initial profile, we show that KPZ fluctuations occur whenever $\alpha \in (0,1/3]$. In the particular regime $\alpha =1/3$, these KPZ fluctuations suddenly vanish at a deterministic time.