## 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.