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
and polynomially if
The critical percolation value is
for site percolation, and
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
, the percolation clusters conditioned to have size
should converge toward the stable map of parameter
introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.