René Conijn

Abstract: Consider an n x n-box in the triangular lattice. The asymptotic behaviour, as n tends to infinity, of the largest percolation clusters in this box was well studied by Borgs, Chayes, Kesten and Spencer in (1999 and 2001). However some questions remained open. If we restrict ourself to critical percolation the size of the largest cluster is of the order n^(91/48). The first natural question is: does there exist a limiting distribution for the size of the largest cluster scaled by its order? In this talk we discuss the existence of the limiting distribution and introduce conformal measure ensembles as a key ingredient. Furthermore we will see an interesting application of these measure ensembles to the FK-Ising model. Based on joint work with Rob van den Berg, Federico Camia and Demeter Kiss.