arXiv:2611.05516v1 Announce Type: new Abstract: The transition from rough to frozen phases in models of random surfaces is expected to be governed by the Pokrovsky-Talapov law, which predicts a $\varepsilon^{3/2}$-type singularity of the macroscopic height profile (limit shape), as the distance $\varepsilon$ from the frozen phase (facet) is sent to zero. This has been proved so far only in exactly solvable cases, notably the classical, determinantal, 2D dimer model. In this paper we prove the validity of the Pokrovsky-Talapov law for a non-integrable 2D dimer model on the hexagonal lattice, whose Gibbs weight depends upon a weak plaquette interaction among dimers. Our proof is based on a multiscale analysis (constructive fermionic Renormalization Group), which allows us to identify two different regimes with distinct scaling properties and to explain the universality of the Pokrovsky-Talapov exponent $3/2$, in contrast with the interaction-dependent critical exponents characterizing the rough phase. The first regime, corresponding to smaller distances, is characterized by an effective quadratic dispersion relation, while the second comes with an effective linear dispersion relation, typical of Luttinger liquid behavior. As a byproduct of our construction, we control the scaling limit of the interacting dimer model uniformly in the distance from the frozen phase.
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