Title: The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity,
Abstract: We prove that the Tutte embeddings (a.k.a. harmonic/embeddings) of certain
random planar maps converge to $\gamma$-Liouville quantum gravity
($\gamma$-LQG). Specifically, we treat mated-CRT maps, which are discretized
matings of correlated continuum random trees, and $\gamma$ 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
SLE$_{\kappa}$ for $\kappa =16/\gamma^2$ (in the annealed sense) and that
simple random walk on the embedded map converges to Brownian motion (in the
quenched sense). Our arguments also yield analogous statements for the Smith
(square tiling) embedding of the mated-CRT map.
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 (using 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.