A Boltzmann approach to percolation on random triangulations

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, 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 root 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 7/6 introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.