The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

Ewain Gwynne, Jason Miller, Scott Sheffield

Published 2017-05-31Version 1

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embedding) of a certain class of random planar maps converge to Liouville quantum gravity (LQG). Specifically, we treat the mated-CRT map, which is a discrete version of the mating of two correlated continuum random trees, and the LQG parameter $\gamma$ ranges from $0$ to $2$ as one varies the correlation parameter. We also show that the associated space-filling path converges under this embedding to space-filling SLE and that random walk on the embedded map converges to Brownian motion. Several recent papers have shown that random planar maps converge to SLE-decorated LQG in important ways (as path-decorated metric spaces, as mated pairs of trees, as collections of loop interfaces, etc.) but this paper is the first to prove that discrete conformal embeddings of random planar maps approximate their continuum counterparts. The main technical contribution is a robust suite of random walk in random environment techniques that apply to random environments on random surfaces, and that we expect to be useful in other settings as well. These techniques still apply, for example, if the edges of the mated-CRT map are randomly weighted in various ways, or if other local modifications to the graph are made.