Entropic repulsion in $|\nabla \phi|^p$ surfaces: a large deviation bound for all $p\geq 1$

We consider the $(2+1)$-dimensional generalized solid-on-solid (SOS) model, that is the random discrete surface with a gradient potential of the form $|\nabla\phi|^{p}$, where $p\in [1,+\infty]$. We show that at low temperature, for a square region $\Lambda$ with side $L$, both under the infinite volume measure and under the measure with zero boundary conditions around $\Lambda$, the probability that the surface is nonnegative in $\Lambda$ behaves like $\exp(-4\beta\tau_{p,\beta} L H_p(L) )$, where $\beta$ is the inverse temperature, $\tau_{p,\beta}$ is the surface tension at zero tilt, or step free energy, and $H_p(L)$ is the entropic repulsion height, that is the typical height of the field when a positivity constraint is imposed. This generalizes recent results obtained in \cite{CMT} for the standard SOS model ($p=1$).