Critical percolation with many interfaces

Percolation is the study of the connections properties of random graphs. One of the most important model of random graph is the so called Bernoulli percolation. In this model, one take a graph (in general the square lattice) consider each edge sequentially and decide to remove it with probability 1-p independently. This random graph exhibit a phase transition at a certain parameter called the critical probability (for the square lattice the critical value is 1/2).

The probability that the centre of a large box of the square lattice is connected to the boundary of the same box is well understood. This correspond to the probability that the centre of a large porous stone is wetted when the stone is immersed in water. For my thesis I plan to study this probability of connection of the centre in systems conditioned to have a large number of connections, in the critical phase. If I can do this then I could try to extend the result to heigh functions models. This would prove some non dependence of random fields with respect to their boundary conditions.