# 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}, journal={Canadian Journal of Mathematics}, year={2017}, volume={71}, pages={1 - 43} }

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…

## 17 Citations

### Percolation probability and critical exponents for site percolation on the UIPT

- MathematicsCanadian Journal of Mathematics
- 2022

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…

### Duality of random planar maps via percolation

- 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…

### Infinite random planar maps related to Cauchy processes

- 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…

### Brownian limits of planar maps with a prescribed degree sequence

- Mathematics
- 2019

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…

### We call a random triangulation distributed according to this limiting law the Infinite Ising

- 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…

### Simple peeling of planar maps with application to site percolation

- MathematicsCanadian Journal of 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…

### Limits of the boundary of random planar maps

- Mathematics
- 2017

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…

### Local convergence of large random triangulations coupled with an Ising model

- MathematicsTransactions 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…

### Scaling limits of random bipartite planar maps with a prescribed degree sequence

- MathematicsRandom Struct. Algorithms
- 2018

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.

### The geometry of a critical percolation cluster on the UIPT

- MathematicsAnnales 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…

## References

SHOWING 1-10 OF 30 REFERENCES

### Percolation on random triangulations and stable looptrees

- Mathematics
- 2013

We study site percolation on Angel and Schramm’s uniform infinite planar triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of…

### Percolation on uniform infinite planar maps

- Mathematics
- 2014

We construct the uniform infinite planar map (UIPM), obtained as the $n \to \infty$ local limit of planar maps with $n$ edges, chosen uniformly at random. We then describe how the UIPM can be sampled…

### Growth and percolation on the uniform infinite planar triangulation

- Mathematics
- 2002

AbstractA construction as a growth process for sampling of the uniform in-
finite planar triangulation (UIPT), defined in [AnS], is given. The
construction is algorithmic in nature, and is an…

### Universal aspects of critical percolation on random half-planar maps

- Mathematics
- 2014

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…

### Percolations on random maps I: Half-plane models

- Mathematics
- 2013

We study Bernoulli percolations on random lattices of the half-plane obtained as local limit of uniform planar triangulations or quadrangulations. Using the characteristic spatial Markov property or…

### Invariance principles for random bipartite planar maps

- Mathematics
- 2007

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…

### Limits of the boundary of random planar maps

- Mathematics
- 2017

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…

### Random planar maps & growth-fragmentations

- Mathematics
- 2015

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…

### The geometry of a critical percolation cluster on the UIPT

- MathematicsAnnales 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…

### The Peeling Process of Infinite Boltzmann Planar Maps

- MathematicsElectron. J. Comb.
- 2016

It is shown that the peeling process on the infinite Boltzmann planar map can be obtained from thepeeling process of finite random maps by conditioning the perimeter process to stay positive, and the scaling limit of the associated perimeter and volume process for arbitrary regular critical weight sequences is obtained.