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 treeweighted maps, bipolaroriented maps, and matedCRT mapstypically 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 matedCRT map together with the relationship of the latter map to SLEdecorated LQG.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 DOI:
 10.48550/arXiv.1807.01512
 arXiv:
 arXiv:1807.01512
 Bibcode:
 2018arXiv180701512G
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 43 pages, 4 figures