  We study Bernoulli percolations on random lattices of the half-plane obtained
as local limit of uniform planar triangulations or quadrangulations. Using the
characteristic spatial Markov property or peeling process of these random
lattices we prove a surprisingly simple universal formula for the critical
threshold for bond and face percolations on these graphs. Our techniques also
permit us to compute off-critical and critical exponents related to percolation
clusters such as the volume and the perimeter.
