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

Dmitry Chelkak

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.

http://scgp.stonybrook.edu/video_portal/video.php?id=4479

Simons- Program: Renormalization and universality in Conformal Geometry, Dynamics, Random Processes, and Field Theory