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A Boltzmann Approach to Percolation on Random Triangulations

Part of: Special processes Geometric probability and stochastic geometry Combinatorics

Published online by Cambridge University Press:  07 January 2019

Olivier Bernardi ,
Nicolas Curien  and
Grégory Miermont
Olivier Bernardi
Affiliation:
Department of Mathematics, Brandeis University, USA Email: bernardi@brandeis.edu
Nicolas Curien
Affiliation:
Département de Mathématiques de l’Université Paris-Sud, and Institut Universitaire de France Email: nicolas.curien@gmail.com
Grégory Miermont
Affiliation:
Unité de Mathématiques Pures et Appliquées de l’École Normale Supérieure de Lyon, and Institut Universitaire de France Email: gregory.miermont@ens-lyon.fr
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Abstract

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We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length $n$ decays exponentially with $n$ except at a particular value $p_{c}$ of the percolation parameter $p$ for which the decay is polynomial (of order $n^{-10/3}$). Moreover, the probability that the origin cluster has size $n$ decays exponentially if $p<p_{c}$ and polynomially if $p\geqslant p_{c}$.

The critical percolation value is $p_{c}=1/2$ for site percolation, and $p_{c}=(2\sqrt{3}-1)/11$ for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.

Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at $p_{c}$, the percolation clusters conditioned to have size $n$ should converge toward the stable map of parameter $\frac{7}{6}$ introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

Keywords

random mapstable mapcritical percolationgasket

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60D05: Geometric probability and stochastic geometry 05A16: Asymptotic enumeration
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 1 , February 2019 , pp. 1 - 43
Copyright
© Canadian Mathematical Society 2018 

Footnotes

We thank the Newton institute for hospitality during the Random Geometry program in 2015 where part of this work was completed. We acknowledge the support of the NSF grant DMS-1400859, and of the Agence Nationale de la Recherche via the grants ANR Liouville (ANR-15-CE40-0013) and ANR GRAAL (ANR-14-CE25-0014).

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