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

Abstract

We introduce a general technique for proving estimates for certain random planar maps which belong to the γ-Liouville quantum gravity (LQG) universality class for γ ∈ (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; γ = 8/3); and planar maps weighted by the number of spanning trees (γ = √ 2), bipolar orientations (γ = 4/3) or Schnyder woods (γ = 1) they admit.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 γ-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 γ = 8/3, we instead deduce estimates for the 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.

The Fractal Dimension of Liouville Quantum Gravity: Universality, Monotonicity, and Bounds

“…This allows us to transfer Theorem 1.6 from the case of the mated-CRT map to the case of these other maps. We do not need to use the bijections mentioned above directly: rather, we will just cite results from [GHS17].…”

“…We first show that the diameter (in the adjacency graph) of the set of -mass cells in the SLE/LQG representation of the mated-CRT map which intersect the Euclidean unit ball is of order −1/dγ +o (1) with high probability (Proposition 4.7), using the comparison results of the preceding subsection and the bounds for Liouville graph distance from Theorem 1.4. We then use this to show that the volume of the graph distance ball of radius r in the mated-CRT map is of order r dγ +or(1) (essentially by taking = 1/r dγ ), and finally transfer to other planar maps using the coupling results of [GHS17].…”

Harmonic functions on mated-CRT maps

“…The reason for this is that in many cases it is possible to transfer estimates from the mated-CRT map to estimates for other random planar maps modulo polylogarithmic multiplicative errors. So far, this has been done for graph distances [GHS17], random walk speed [GM17,GH18], and random walk return probabilities [GM17]. However, we have not yet found a way to transfer modulus of continuity bounds for harmonic functions, which is what is needed to deduce an analog of Corollary 1.6 for other planar map models.…”

Subsequential Scaling Limits for Liouville Graph Distance

“…Note that in these works there is no mention of a discrete approximation of the metric. In [14,9,20,19,10], various bounds on the distance and properties of the geodesics were obtained. In [13,11], some non-universality aspects (when considering underlying logcorrelated fields other than GFF) for LQG distances were demonstrated.…”