Bipartite dimer model: Gaussian Free Field on Lorentz-minimal surfaces

Abstract: We discuss a new viewpoint on the convergence of fluctuations in the bipartite dimer model considered on big planar graphs. Classically, when these graphs are parts of refining lattices, the boundary profile of the height function and a lattice-dependent entropy functional are responsible for the conformal structure, in which the limiting GFF (and CLE(4)) should be defined. Motivated by a long-term perspective of understanding the `discrete conformal structure’ of random planar maps equipped with the dimer (or the critical Ising) model, we introduce `perfect t-embeddings’ of abstract weighted bipartite graphs and argue that such embeddings reveal the conformal structure in a universal way: as that of a related Lorentz-minimal surface in 2+1 (or 2+2) dimensions.

Though the whole concept is very new, concrete deterministic examples (e.g, the Aztec diamond) justify its relevance, and general convergence theorems obtained so far are of their own interest. Still, many open questions remain, one of the key ones being to understand the mechanism behind the appearance of the Lorentz metric in this classical problem.

Based upon recent joint works with Benoît Laslier, Sanjay Ramassamy and Marianna Russkikh.

dynamical systemsprobability

Audience: researchers in the topic

( slides | video )

Horowitz seminar on probability, ergodic theory and dynamical systems