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Percolation probability and critical exponents for site percolation on the UIPT

Part of: Special processes Graph theory Combinatorics Combinatorial probability Geometric probability and stochastic geometry

Published online by Cambridge University Press:  20 October 2022

Laurent Ménard
Laurent Ménard*
Affiliation:
New York University Shanghai, Shanghai, China and Laboratoire Modal’X, Université Paris Nanterre, Nanterre, France
*
e-mail: laurent.menard@normalesup.org
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Abstract

We derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $\beta = 1/2$. Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like $n^{-1/7}$. Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like $n^{-4/3}$. Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.

Keywords

Random planar mapstriangulationsUIPTpercolationcritical exponents

MSC classification

Primary: 05A15: Exact enumeration problems, generating functions 05A16: Asymptotic enumeration 05C12: Distance in graphs 05C30: Enumeration in graph theory 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 6 , December 2023 , pp. 1869 - 1903
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work is partially supported by the ANR grant ProGraM (Projet-ANR-19-CE40-0025) and the Labex MME-DII (ANR11-LBX-0023-01).

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