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.