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 sitepercolation or bondpercolation) 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 $\sqrt$ 31 11 for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for sitepercolation, and by Angel \& Curien for bondpercolation, 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 arXiv:
 arXiv:1705.04064
 Bibcode:
 2017arXiv170504064B
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Probability
 EPrint:
 Can. J. Math.J. Can. Math. 71 (2019) 143