also associate an interface growth model to our microscopic model, roughly speaking by integrating it in the space variable, and show that it converges to the energy solution of the KPZ equation, thereby giving a physical justification of the Neumann boundary conditions.

In the remaining lines we go into further details and explain these results.

### WASEP with reservoirs

At the microscopic level, the model that we consider is the following: let a discrete system of particles evolve on the finite set $\Lambda_n=\{1,\dots,n-1\}$ of size $n\in{{\mathbb N}}$. For any site $x\in\Lambda_n$ there is at most one particle, and we denote by $\eta(x)\in\{0,1\}$ its occupation variable, namely: $\eta(x)=1$ if there is a particle at site $x$, and 0 otherwise. We then define a continuous-time Markov process $\eta_t=\{\eta_t(x)\; ; \; x\in\Lambda_n\}$ on the state space $\{0,1\}^{\Lambda_n}$ using the following possible moves: for any $x \notin\{1,n-1\}$, a particle at site $x$ can attempt to jump to its neighbouring sites $x-1$ or $x+1$, provided that they are empty. Similarly, a particle at site $1$ can jump to its right neighbour $2$ or it can leave the system, and the particle at site $n-1$ can jump to its left neighbour $n-2$ or it can leave 