A central limit theorem for square ice

Wu, Wei

doi:10.48550/arxiv.2206.12058

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2022

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“…Let us mention in particular the recent [36], which proves with a rather soft argument a (non-quantitative) delocalization statement for rather general height models, under the restriction, however, that the underlying graph has maximal degree three. For the particular case of the 6-vertex model, delocalization of the height function is known to hold in several regions of parameters [14,15,38,42] but full scaling to the GFF has been proven only in a neighborhood of the free fermion point [27].…”

Section: Introductionmentioning

confidence: 99%

Weakly non-planar dimers

Giuliani¹,

Bruno²,

Toninelli³

2022

Preprint

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We study a model of fully-packed dimer configurations (or perfect matchings) on a bipartite periodic graph that is two-dimensional but not planar. The graph is obtained from Z 2 via the addition of an extensive number of extra edges that break planarity (but not bipartiteness). We prove that, if the weight λ of the non-planar edges is small enough, the height function scales on large distances to the Gaussian Free Field with a λ-dependent amplitude, that coincides with the anomalous exponent of dimer-dimer correlations. Because of non-planarity, Kasteleyn's theory does not apply: the model is non-determinantal. Rather, we map the model to a system of interacting lattice fermions in the Luttinger universality class, that we then analyze via fermionic Renormalization Group methods.

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