Entropic repulsion $$|\nabla \phi |^p$$ surfaces: a large deviation bound for p>1 | ∇ ϕ | p surfaces: a large deviation bound for all $$p\ge 1$$ p ≥ 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 |∇φ|p, where p ∈ [1, +∞]. We show that at low temperature, for a square region with side L , both under the infinite volume measure and under the measure with zero boundary conditions around , the probability that the surface is nonnegative in behaves like exp(−4βτp,β L Hp(L)), where β is the inverse temperature, τp,β is the surface tension at zero tilt, or step free energy, and Hp(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 [8] for the standard SOS model (p = 1).