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.