  There are many classical random walk in random environment results that apply
to ergodic random planar environments. We extend some of these results to
random environments in which the length scale varies from place to place, so
that the law of the environment is in a certain sense only translation
invariant {\em modulo scaling}. For our purposes, an ``environment'' consists
of an infinite random planar map embedded in $\mathbb C$, each of whose edges
comes with a positive real conductance. Our main result is that under modest
constraints (translation invariance modulo scaling together with the finiteness
of a type of specific energy) a random walk in this kind of environment
converges to Brownian motion modulo time parameterization in the quenched
sense.
  Environments of the type considered here arise naturally in the study of
random planar maps and Liouville quantum gravity. In fact, the results of this
paper are used in separate works to prove that certain random planar maps
(embedded in the plane via the so-called Tutte embedding) have scaling limits
given by SLE-decorated Liouville quantum gravity, and also to provide a more
explicit construction of Brownian motion on the Brownian map. However, the
results of this paper are much more general and can be read independently of
that program.
  One general consequence of our main result is that if a translation invariant
(modulo scaling) random embedded planar map and its dual have finite energy per
area, then they are close on large scales to a minimal energy embedding (the
harmonic embedding). To establish Brownian motion convergence for an {\em
infinite} energy embedding, it suffices to show that one can perturb it to make
the energy finite.
