A Boltzmann Approach to Percolation on Random Triangulations

Bernardi, Olivier ; Curien, Nicolas ; Miermont, Grégory

Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 1-43

Résumé

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 $p\geqslant p_{c}$.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 triangulations identified by Angel for site-percolation, and by Angel and 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 $\frac{7}{6}$ introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

Détail

DOI : 10.4153/CJM-2018-009-x

Mots-clés : random map, stable map, critical percolation, gasket