Vol. 2, No. 1, 2021

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Infinite stable Boltzmann planar maps are subdiffusive

Nicolas Curien and Cyril Marzouk

Vol. 2 (2021), No. 1, 1–26
Abstract

The infinite discrete stable Boltzmann maps are generalisations of the well-known uniform infinite planar quadrangulation in the case where large degree faces are allowed. We show that the simple random walk on these random lattices is always subdiffusive with exponent less than 1 3. Our method is based on stationarity and geometric estimates obtained via the peeling process which are of individual interest.

Keywords
random maps, random walk, subdiffusivity, anomalous diffusion, peeling
Mathematical Subject Classification 2010
Primary: 05C81, 60D05
Secondary: 05C80, 60G10, 82B41
Milestones
Received: 29 October 2019
Revised: 28 August 2020
Accepted: 12 October 2020
Published: 16 March 2021
Authors
Nicolas Curien
Département de Mathématique
Université Paris–Saclay
Faculté des Sciences d’Orsay
Orsay
France
Cyril Marzouk
Centre de Mathématiques Appliquées
École Polytechnique
Palaiseau
France