A Boltzmann approach to percolation on random triangulations

Olivier Bernardi (1) , Nicolas Curien (2) , Grégory Miermont (3)

1 Brandeis University
2 Université Paris-Saclay
3 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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 ≥ p_c. The critical percolation value is p_c = 1/2 for site percolation, and p c = 2 √ 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.

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]

Olivier Bernardi, Nicolas Curien, Grégory Miermont. A Boltzmann approach to percolation on random triangulations. Canadian Journal of Mathematics, 2019. ⟨hal-01517947v3⟩