Abstract
We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most $$n^{1/4 + o_n(1)}$$
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 $$n^{1/4 + o_n(1)}$$
n
1
/
4
+
o
n
(
1
)
, as conjectured by Benjamini and Curien (Geom Funct Anal 2(2):501–531, 2013. arXiv:1202.5454). More generally, we show that the simple random walks on a certain family of random planar maps in the $$\gamma $$
γ
-Liouville quantum gravity (LQG) universality class for $$\gamma \in (0,2)$$
γ
∈
(
0
,
2
)
—including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps—typically travels graph distance $$n^{1/d_\gamma + o_n(1)}$$
n
1
/
d
γ
+
o
n
(
1
)
in n units of time, where $$d_\gamma $$
d
γ
is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on $$\gamma $$
γ
by Ding and Gwynne (Commun Math Phys 374:1877–1934, 2018. arXiv:1807.01072). Since $$d_\gamma > 2$$
d
γ
>
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 $${\mathbb {C}}$$
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.