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Anomalous diffusion of random walk on random planar maps

Gwynne, Ewain and Hutchcroft, Tom (2020) Anomalous diffusion of random walk on random planar maps. Probability Theory and Related Fields, 178 (1-2). pp. 567-611. ISSN 0178-8051. doi:10.1007/s00440-020-00986-7.

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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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Hutchcroft, Tom0000-0003-0061-593X
Additional Information:© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit Received: 5 September 2018 / Revised: 21 May 2020 / Published online: 20 July 2020. We thank two anonymous referees for helpful comments on an earlier version of this manuscript. We thank Marie Albenque, Nina Holden, Jason Miller, Asaf Nachmias, and Xin Sun for helpful discussions. We thank Asaf in particular for bringing the maximal versions of the Markov-type inequalities to our attention. This work was initiated during a visit by TH to MIT, whom he thanks for their hospitality.
Issue or Number:1-2
Classification Code:Mathematics Subject Classification Primary 60K50 (Anomalous diffusion models) Secondary 60J67 (SLE) 60D05 (geometric probability)
Record Number:CaltechAUTHORS:20210922-193307816
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Official Citation:Gwynne, E., Hutchcroft, T. Anomalous diffusion of random walk on random planar maps. Probab. Theory Relat. Fields 178, 567–611 (2020).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110995
Deposited By: George Porter
Deposited On:27 Sep 2021 20:43
Last Modified:12 Oct 2022 15:37

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