Tilted Solid-On-Solid is liquid: scaling limit of SOS with a potential on a slope

The $(2+1)$D Solid-On-Solid (SOS) model famously exhibits a roughening transition: on an $N\times N$ torus with the height at the origin rooted at $0$, the variance of $h(x)$, the height at $x$, is $O(1)$ at large inverse-temperature $\beta$, vs. $\asymp \log |x|$ at small $\beta$ (as in the Gaussian free field (GFF)). The former--rigidity at large $\beta$--is known for a wide class of $|\nabla\phi|^p$ models ($p=1$ being SOS) yet is believed to fail once the surface is on a slope (tilted boundary conditions). It is conjectured that the slope would destabilize the rigidity and induce the GFF-type behavior of the surface at small $\beta$. The only rigorous result on this is by Sheffield (2005): for these models of integer height functions, if the slope $\theta$ is irrational, then Var$(h(x))\to\infty$ with $|x|$ (with no known quantitative bound). We study a family of SOS surfaces at a large enough fixed $\beta$, on an $N\times N$ torus with a nonzero boundary condition slope $\theta$, perturbed by a potential $V$ of strength $\epsilon_\beta$ per site (arbitrarily small). Our main result is (a) the measure on the height gradients $\nabla h$ has a weak limit $\mu_\infty$ as $N\to\infty$; and (b) the scaling limit of a sample from $\mu_\infty$ converges to a full plane GFF. In particular, we recover the asymptotics Var$(h(x))\sim c\log|x|$. To our knowledge, this is the first example of a tilted $|\nabla\phi|^p$ model, or a perturbation thereof, where the limit is recovered at large $\beta$. 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) leads to the GFF limit. New ingredients are needed in both parts, including a nontrivial extension of [GMT17] from finite interactions to any long range summable interactions.