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Printer-friendly versionFriday, April 19, 2024
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04/19/2024 - 10:00am 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. Location:
KT801
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