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+on)(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+on) (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-Liouville quantum gravity (LQG) universality class for gamma is an element of (0, 2)-including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps-typically travels graph distance n(1/d gamma+on) (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 (Commun Math Phys 374:1877-1934, 2018. arXiv:1807.01072). 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 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.