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Perfect t-embeddings of bipartite planar graphs and the convergence to the GFF -I Laslier, Benoit


We discuss a concept of `perfect t-embeddingsâ , or `p-embeddings', of weighted bipartite planar graphs. (T-embeddings also appeared under the name Coulomb gauges in a recent work of Kenyon, Lam, Ramassamy and Russkikh.) We believe that these p-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. To support this idea, we first develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon's holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. Further, given a sequence of (abstract) planar graphs G_n and their p-embeddings T_n onto the unit disc D, assume that (i) the faces of T_n satisfy certain technical assumptions in the bulk of D; (ii) the size of the associated origami maps O_n tends to zero as n grows (again, on each compact subset of D). We prove that (i)+(ii) imply the convergence of the fluctuations of the dimer height functions on G_n (provided that these graphs are embedded by T_n), to the GFF on the unit disc D equipped with the standard complex structure. Though this is not fully clear at the moment, we conjecture that the origami maps O_n are always small in absence of frozen regions and gaseous bubbles, so our theorem can be eventually applied to all such cases. Moreover, the same techniques are applicable in the situation when the limit of the origami maps arising from a sequence of p-embeddings is a Lorenz-minimal surface, in this situation one eventually obtains the GFF in the conformal parametrization of this surface. In a related joint work with Sanjay Ramassamy we argue that such a Lorenz-minimal surface indeed arises in the case of classical Aztec diamonds; a general conjecture is that this should `always' be the case due to a link between p-embeddings and a representation of the dimer model in the Plucker quadric. Time permitting, we also indicate how the theory of t-holomorphic functions specifies to the Ising case and discuss related results on conformal invariance of the Ising model as well as a more general perspective.

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