Weakly asymmetric bridges and the KPZ equation

We consider a discrete bridge from (0, 0) to (2N, 0) evolving according to the corner growth dynamics, where the jump rates are subject to an upward asymmetry of order N with α > 0. We provide a classification of the static and dynamic behaviour of this model according to the value of the parameter α. Our main results concern the hydrodynamic limit and the fluctuations of the bridge. For α < 1, the hydrodynamic limit is given by the entropy solution of the inviscid Burgers equation with Dirichlet boundary conditions. For α ≤ 1/3, we show that the fluctuations around this hydrodynamic limit are given by the KPZ equation on the line and restricted to the time interval [0, T ): the final time T is infinite when α < 1/3, while in the regime α = 1/3 it equals the finite time needed by the hydrodynamic limit to reach its stationary state.