Anomalous diffusion of random walk on random planar maps

Abstract

We prove that the simple random walk on the uniform infinite planar triangulation (UIPT) typically travels graph distance at most $n^{1/4 + o_n(1)}$ 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.