Anomalous diffusion of random walk on random planar maps

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.