A mating-of-trees approach for graph distances in random planar maps

TL;DR

This paper introduces a new technique for estimating graph distances in a broad class of random planar maps within the $ ext{LQG}$ universality class, using a mating-of-trees approach and strong coupling with mated-CRT maps.

Contribution

The paper develops a general mating-of-trees technique to derive bounds on graph distances in various random planar maps, connecting discrete models with continuum theory without relying on SLE/LQG.

Findings

Established bounds for graph distances consistent with Watabiki's prediction.

Proved the existence of an exponent for certain distances in these maps.

Connected discrete random maps to continuum models via strong coupling.

Abstract

We introduce a general technique for proving estimates for certain random planar maps which belong to the $\gamma$-Liouville quantum gravity (LQG) universality class for $\gamma \in (0,2)$. The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d.\ increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; $\gamma=\sqrt{8/3}$); and planar maps weighted by the number of different spanning trees ($\gamma=\sqrt 2$), bipolar orientations ($\gamma=\sqrt{4/3}$), or Schnyder woods ($\gamma=1$) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (1993) prediction for the Hausdorff dimension of $\gamma$-LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map $M$ to a mated-CRT map---a random planar map constructed from a correlated two-dimensional Brownian motion---using a strong coupling (Zaitsev, 1998) of the encoding walk for $M$ and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in $M$ from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when $\gamma=\sqrt{8/3}$, we instead deduce estimates for the $\sqrt{8/3}$-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.