We consider a class of random interface models on the one-dimensional discrete torus $T_N$ parametrised by a positive map $\Phi_N$ on $T_N\times\mathbb{R}$. These models share a weak perturbation, namely an asymmetry of order $N^{-\gamma}$ of the direction of growth that switches from up to down and is always towards the direction that reduces the size of the difference of $\Phi_N$ above and below the interface. We specialise to the case of constant $\Phi_N$, so that the asymmetry direction is based on the sign of the area underneath the interface, and study the hydrodynamic limit, stationary correlation functions and equilibrium fluctuations of the interface. We will also discuss the case of a more general $\Phi_N$. Based on joint work with Martin Hairer and Patrícia Gonçalves.