[Paper Review] A mating-of-trees approach for graph distances in random planar maps

Ewain Gwynne, Nina Holden+1 more|arXiv (Cornell University)|Nov 2, 2017

Stochastic processes and statistical mechanics56 references11 citations

TL;DR

This paper introduces a mating-of-trees coupling technique to derive non-trivial upper and lower bounds for graph distance balls in random planar maps belonging to the γ-Liouville quantum gravity (LQG) universality class for γ ∈ (0, 2). By strongly coupling the encoding random walk of such maps to a correlated Brownian motion, the authors transfer distance estimates from the mated-CRT map—previously established via continuum theory—to discrete random planar maps, yielding the first non-trivial bounds for maps weighted by spanning trees, bipolar orientations, and Schnyder woods, and sharper bounds for the p8/3-mated-CRT map.

ABSTRACT

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 different spanning trees (γ=2), bipolar orientations (γ=4/3), or Schnyder woods (γ=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 (Prog Theor Phys Suppl 114:1–17, 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 in ESAIM Probab Stat 2:41–108, 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.

Motivation & Objective

• To develop a general framework for estimating graph distances in random planar maps within the γ-Liouville quantum gravity (LQG) universality class.
• To extend distance estimates from the mated-CRT map—constructed from correlated Brownian motion—to discrete random planar maps via strong coupling.
• To provide the first non-trivial bounds on the cardinality of graph distance balls for maps weighted by spanning trees, bipolar orientations, and Schnyder woods.
• To refine distance estimates for the p8/3-mated-CRT map, improving upon prior results.

Proposed method

• Use a strong coupling (Zaitsev, 1998) between the encoding random walk of a planar map and a correlated two-dimensional Brownian motion.
• Leverage the mated-CRT map as a continuum proxy, constructed from the Brownian motion, for which graph distance estimates were previously established via continuum theory.
• Apply the coupling to show that the mated-CRT map and the discrete map are roughly isometric up to a polylogarithmic factor with high probability.
• Use the bijection between planar maps and random walks (mating-of-trees) to encode maps with i.i.d. increment walks.
• Treat specific map models—UIPT, spanning-tree-decorated, bipolar-oriented, and Schnyder-wood-decorated maps—individually using their respective bijections.
• Transfer distance estimates from the mated-CRT map to the discrete map using the coupling, relying on the fact that the coupling preserves metric structure up to logarithmic factors.

Experimental results

Research questions

• RQ1 Can non-trivial bounds for graph distance balls be established in random planar maps weighted by spanning trees, bipolar orientations, or Schnyder woods?
• RQ2 Does the mated-CRT map serve as a robust proxy for discrete random planar maps in terms of graph distance scaling?
• RQ3 Can the Watabiki (1993) prediction for the Hausdorff dimension of γ-LQG be confirmed via discrete estimates in the γ-LQG universality class?
• RQ4 What is the scaling exponent χ for distances in γ-LQG random planar maps, and does it correspond to the Gromov-Hausdorff scaling limit?
• RQ5 Can the results for the p8/3-mated-CRT map be improved beyond existing bounds in [GHS19]?

Key findings

• The paper establishes non-trivial upper and lower bounds for the cardinality of graph distance balls in γ-LQG random planar maps, consistent with Watabiki's prediction for the Hausdorff dimension.
• It proves the existence of a scaling exponent χ for distances in the planar map, which is expected to govern the Gromov-Hausdorff scaling limit for γ ≤ √2.
• For the p8/3-mated-CRT map, the paper obtains sharper distance bounds than those in [GHS19], improving the state of the art.
• The first non-trivial bounds for graph distances are derived for random planar maps weighted by spanning trees (γ = √2), bipolar orientations (γ = p4/3), and Schnyder woods (γ = 1).
• The coupling technique enables transfer of metric estimates from the mated-CRT map to discrete maps with high probability, up to a polylogarithmic factor.
• The results are robust and extendable to any map model encoded by a mating-of-trees bijection with i.i.d. increment walks, suggesting broad applicability.