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Simple peeling of planar maps with application to site percolation

Part of: Graph theory Special processes

Published online by Cambridge University Press:  26 February 2021

Timothy Budd  and
Nicolas Curien
Timothy Budd*
Affiliation:
Institute for Mathematics, Astrophysics and Particle Physics, Faculty of Science, Radboud University, Nijmegen, The Netherlands
Nicolas Curien
Affiliation:
Université Paris-Saclay, Orsay, France and Institut Universitaire de France, Paris, France e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolations on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations, distinct vertices on the exploration contour may, at latter stage, be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative “simple” peeling exploration of Angel and uncover deep relations with the lazy-peeling process. Along the way, we define and study the random Boltzmann map of the half-plane with a simple boundary for an arbitrary critical weight sequence. Its construction is nontrivial especially in the “dense regime,” where the half-planar random Boltzmann map does not possess an infinite simple core.

Keywords

Random planar mapspeeling processsimple boundaryBernoulli site percolation

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 907 - 941
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first author acknowledges support of the START-UP 2018 programme with project number 740.018.017, which is financed by the Dutch Research Council (NWO). The second author is supported by ERC GeoBrown (ERC Advanced Grant 740943).

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