• DOI:10.1007/s00440-020-00986-7
• Corpus ID: 119151878

Anomalous diffusion of random walk on random planar maps

@article{Gwynne2020AnomalousDO,
  title={Anomalous diffusion of random walk on random planar maps},
  author={Ewain Gwynne and Tom Hutchcroft},
  journal={Probability Theory and Related Fields},
  year={2020}
}

• Ewain Gwynne, Tom Hutchcroft
• Published 4 July 2018
• Mathematics
• Probability Theory and Related Fields

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.