# A Boltzmann approach to percolation on random triangulations

## TL;DR

This paper investigates percolation on Boltzmann triangulations using generating functions, identifying phase transitions and critical thresholds that match known results, and explores the geometry of percolation clusters at criticality.

## Contribution

It provides an independent derivation of percolation thresholds on Boltzmann triangulations and connects criticality conditions to stable maps, offering new insights into critical exponents.

## Key findings

- Critical percolation threshold for site percolation is p_c=1/2.
- Critical percolation threshold for bond percolation is p_c=2√3−1.
- Percolation interface length decay shifts from exponential to polynomial at p_c.

## 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 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 $\ge$ 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 triangula-tions identified by Angel for site-percolation, and by Angel \& 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 7 6 introduced by Le Gall \& Miermont. This enables us to derive heuristically some new critical exponents.