# A Boltzmann Approach to Percolation on Random Triangulations

@article{Bernardi2017ABA,
title={A Boltzmann Approach to Percolation on Random Triangulations},
author={Olivier Bernardi and Nicolas Curien and Gr{\'e}gory Miermont},
year={2017},
volume={71},
pages={1 - 43}
}
• Published 11 May 2017
• Mathematics
Abstract 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…
We derive three critical exponents for Bernoulli site percolation on the on the Uniform Infinite Planar Triangulation (UIPT). First we compute explicitly the probability that the root cluster is
• Mathematics, Physics
• 2018
We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\alpha\in(1,2]$. We
• Mathematics
• 2017
We study the geometry of infinite random Boltzmann planar maps having weight of polynomial decay of order $k^{-2}$ for each vertex of degree $k$. These correspond to the dual of the discrete "stable
We consider uniformly random bipartite planar maps with a given boundary-length and $n$ inner face with given degrees and we study its asymptotic behaviour as $n \to \infty$. We prove that, suitably
• Mathematics, Physics
• 2018
We show that the uniform measure on triangulations of size n with an Ising configuration biased by the energy of the configuration converges weakly as n→∞ for the local topology. To do so, for any
• Mathematics
• 2021
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
We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable
• Mathematics
Transactions of the American Mathematical Society
• 2020
We prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size n n decorated by an Ising configuration with a weight proportional to the
This paper studies the asymptotic behaviour of uniform random maps with a prescribed face-degree sequence, in the bipartite case, and shows that, properly rescaled, such maps converge in distribution towards the Brownian map in the Gromov-Hausdorff sense.
• Mathematics
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
• 2018
We consider a critical Bernoulli site percolation on the uniform infinite planar triangulation. We study the tail distributions of the peeling time, perimeter, and volume of the hull of a critical

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We study a large class of Bernoulli percolation models on random lattices of the half- plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact
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It is conjectured in the Physics literature that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not
We discuss asymptotics for the boundary of critical Boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable
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We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these
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We consider a critical Bernoulli site percolation on the uniform infinite planar triangulation. We study the tail distributions of the peeling time, perimeter, and volume of the hull of a critical
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