  We prove that the simple random walk on the uniform infinite planar
triangulation (UIPT) typically travels graph distance at most $n^{1/4 +
o_n(1)}$ in $n$ units of time. Together with the complementary lower bound
proven by Gwynne and Miller (2017) this shows that the typical graph distance
displacement of the walk after $n$ steps is $n^{1/4 + o_n(1)}$, as conjectured
by Benjamini and Curien (2013). More generally, we show that the simple random
walks on a certain family of random planar maps in the $\gamma$-Liouville
quantum gravity (LQG) universality class for $\gamma\in (0,2)$---including
spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT
maps---typically travels graph distance $n^{1/d_\gamma + o_n(1)}$ in $n$ units
of time, where $d_\gamma$ is the growth exponent for the volume of a metric
ball on the map, which was shown to exist and depend only on $\gamma$ by Ding
and Gwynne (2018). Since $d_\gamma > 2$, this shows that the simple random walk
on each of these maps is subdiffusive.
  Our proofs are based on an embedding of the random planar maps under
consideration into $\mathbb C$ wherein graph distance balls can be compared to
Euclidean balls modulo subpolynomial errors. This embedding arises from a
coupling of the given random planar map with a mated-CRT map together with the
relationship of the latter map to SLE-decorated LQG.
