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 γ-Liouville quantum gravity (LQG) universality class for γ∈(0,2)---including spanning tree-weighted maps, bipolar-oriented maps, and mated-CRT maps---typically travels graph distance n^(1/d_γ + o_n(1)) in n units of time, where dγ is the growth exponent for the volume of a metric ball on the map, which was shown to exist and depend only on γ by Ding and Gwynne (2018). Since 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 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.

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