Anomalous diffusion of random walk on random planar maps

TL;DR

This paper proves that simple random walks on certain random planar maps, including the UIPT and models in the LQG universality class, are subdiffusive with displacement growing roughly as n^{1/d_gamma}, using embeddings into the complex plane.

Contribution

It establishes the subdiffusive behavior of random walks on a broad class of random planar maps via novel embeddings and couplings with mated-CRT maps and SLE-decorated LQG.

Findings

Random walk displacement on UIPT is approximately n^{1/4}

Random walks on LQG class maps are subdiffusive with exponent 1/d_gamma

Embedding techniques relate graph and Euclidean distances with subpolynomial errors

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.