Vertex models, random planar maps and Liouville quantum gravity

Xin Sun (MIT)

Abstract :

We explain how to construct spanning trees from two classical vertex models on Euclidean lattice: 6-vertex model and 20-vertex model. The construction can be thought of as a generalization of Temperley bijection between dimer model and uniform spanning trees. It is closely related to Gaussian free field and SLE. Although little is known in the Euclidean case beyond conjectures, we demonstrate that the analog models on random planar maps converge to an independent coupling of imaginary geometry and Liouville quantum gravity in an unconventional but rather strong topology. This line of research is started by Kenyon-Miller-Sheffield-Wilson (2015) and is further developed in a joint work with E. Gwynne and N. Holden and some ongoing projects jointly with Y. Li and S. Watson.