A boltzmann approach to percolation on random triangulations

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 decays exponentially with except at a particular value of the percolation parameter for which the decay is polynomial (of order ). Moreover, the probability that the origin cluster has size decays exponentially if