We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to γ-Liouville quantum gravity (γ-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and γ ranges from 0 to 2 as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling for (in the annealed sense) and that simple random walk on the embedded map converges to Brownian motion (in the quenched sense).
This work constitutes the first proof that a discrete conformal embedding of a random planar map converges to LQG. Many more such statements have been conjectured. Since the mated-CRT map can be viewed as a coarse-grained approximation to other random planar maps (the UIPT, tree-weighted maps, bipolar-oriented maps, etc.), our results indicate a potential approach for proving that embeddings of these maps converge to LQG as well.
To prove the main result, we establish several (independently interesting) theorems about LQG surfaces decorated by space-filling SLE. There is a natural way to use the SLE curve to divide the plane into “cells” corresponding to vertices of the mated-CRT map. We study the law of the shape of the origin-containing cell, in particular proving moments for the ratio of its squared diameter to its area. We also give bounds on the degree of the origin-containing cell and establish a form of ergodicity for the entire configuration. Ultimately, we use these properties to show (with the help of a general theorem proved in a separate paper) that random walk on these cells converges to a time change of Brownian motion, which in turn leads to the Tutte embedding result.
The first author was partially supported by the National Science Foundation Grant DMS-1209044.
The third author was partially supported by the National Science Foundation Grant DMS-1712862 and a Simons Fellowship with award number 306120.
We thank an anonymous referee for helpful comments on an earlier version of this paper. We thank the Mathematical Research Institute of Oberwolfach for its hospitality during a workshop where part of this work was completed.
"The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity." Ann. Probab. 49 (4) 1677 - 1717, July 2021. https://doi.org/10.1214/20-AOP1487