  We study a large class of Bernoulli percolation models on random lattices of
the half- plane, obtained as local limits of uniform planar triangulations or
quadrangulations. We first compute the exact value of the site percolation
threshold in the quadrangular case using the so-called peeling techniques.
Then, we generalize a result of Angel about the scaling limit of crossing
probabilities, that are a natural analogue to Cardy's formula in (non-random)
plane lattices. Our main result is that those probabilities are universal, in
the sense that they do not depend on the percolation model neither on the
degree of the faces of the map.
