Anomalous diffusion of random walk on random planar maps
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Anomalous diffusion of random walk on random planar maps Ewain Gwynne1
· Tom Hutchcroft1
Received: 5 September 2018 / Revised: 21 May 2020 © The Author(s) 2020
Abstract 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 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/dγ +on (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 (Commun Math Phys 374:1877–1934, 2018. arXiv:1807.01072). Since dγ > 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. Mathematics Subject Classification Primary 60K50 (Anomalous diffusion models); Secondary 60J67 (SLE) · 60D05 (geometric probability)
Contents 1 Introduction . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . 1.2 Mated-CRT map background . 1.3 Main result in the general case
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Ewain Gwynne [email protected] University of Cambridge, Cambridge, UK
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E. Gwynne, T. Hutchcroft 1.4 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Perspective and approach . . . . . . . . . . . . . . . . . . . . 1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Unimodular and reversible weighted graphs . . . . . . . . . . 2.2 Markov-type inequalities . . . . . . . . . . . . . . . . . . . . 2.3 Liouville quantum gravity . . . . . . . . . . . . . . . . . . . . 2.3.1 The γ -quantum cone . . . . . . . . . . . . . . . . . . . 2.4 The SLE/LQG description of the mated-CRT map . . . . . . . 2.5 Strong coupling with the mated-CRT map . . . . . . . . . . . 3 The core argument . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Setup . . . . . . . . . . . . . . . . . . .
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