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 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 (Prog Theor Phys Suppl 114:1–17, 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 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 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.