The (2+1)D Solid-On-Solid (SOS) model famously exhibits a roughening transition: on an *N*×*N* torus with the height at the origin rooted at 0, the variance of *h*(*x*), the height at a point *x* is O(1) when the inverse-temperature *β *is large, vs O(log |*x*|) when *β *is small. The rigidity at large *β *is believed to fail once the surface is on a slope (tilted boundary conditions), which ought to destabilize it and induce the log-correlated behavior of the small *β *regine. The only rigorous result on this is by Sheffield (2005): if the slope *θ* is irrational, then Var(*h*(*x*)) diverges with |*x*| (with no known quantitative bound).

We study this model at a large enough fixed *β*, on an *N*×*N* torus with a nonzero boundary condition slope *θ*, perturbed by a potential *V* of strength *ε*(*β)* per site (arbitrarily small). Our main result is (a) the measure on the height gradients ∇*h *has a weak limit *μ* as *N*→∞; and (b) the scaling limit of a sample from *μ *converges to a full plane GFF. In particular, we recover the asymptotics of Var(*h*(*x*)). To our knowledge, this is the first example of a random surface of the ∇*ɸ* family of models, or any perturbation of one, where the scaling limit is recovered at large finite *β *under tilted boundary conditions.

The proof looks at random monotone surfaces that approximate the SOS surface, and shows that (i) these form a weakly interacting dimer model, and (ii) the renormalization framework of Giuliani, Mastropietro and Toninelli (2017) can be applied to it, leading to the scaling limit.

Joint work with Benoît Laslier.