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We derive three critical exponents for Bernoulli site percolation on the onthe Uniform Infinite Planar Triangulation (UIPT). First we compute explicitlythe probability that the root cluster is infinite. As a consequence, we showthat 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 thenumber of vertices in the root cluster. We use this formula to prove that, atcriticality, the probability that the root cluster has at least $n$ verticesdecays like $n^{-1/7}$. Finally, we also derive an expression for the law ofthe 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 $n$ decayslike $n^{-4/3}$. Among these three exponents, only the last one was previouslyknown. Our main tools are the so-called gasket decomposition of percolationclusters, generic properties of random Boltzmann maps, as well as analyticcombinatorics.

Percolation probability and critical exponents for site percolation on the UIPT