  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.
