We derive three critical exponents for Bernoulli site percolation 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, and analytic combinatorics.

This paper addresses the process of estimating loss reserves for a company or syndicate writing in the London Market. Particular emphasis is placed on insurers maximising the value of the process, and ensuring that the process is not simply a series of mathematical calculations. The use of sophisticated mathematical techniques should not distract from the importance of understanding the business and ensuring that data are correct. Sophisticated mathematical techniques can give rise to misleading impressions of confidence and accuracy to estimates, which are often subject to considerable uncertainty. The principles (rather than the detailed techniques) are illustrated by a case study based on a hypothetical London Market writer. Many of these principles are relevant to other markets.