  We derive three critical exponents for Bernoulli site percolation on the on
the Uniform Infinite Planar Triangulation (UIPT). First we compute explicitly
the probability that the root cluster is infinite. As a consequence, we show
that the off-critical exponent for site percolation on the UIPT is $\beta =
1/2$. Then we establish an integral formula for the generating function of the
number of vertices in the root cluster. We use this formula to prove that, at
criticality, the probability that the root cluster has at least $n$ vertices
decays like $n^{-1/7}$. Finally, we also derive an expression for the law of
the perimeter of the root cluster and use it to establish that, at criticality,
the probability that the perimeter of the root cluster is equal to $n$ decays
like $n^{-4/3}$. Among these three exponents, only the last one was previously
known. Our main tools are the so-called gasket decomposition of percolation
clusters, generic properties of random Boltzmann maps, as well as analytic
combinatorics.
