Conformal Invariance in Two-dimensional Percolation

The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it has been in use since the seventeenth century. It was introduced more recently into mathematics by S. R. Broadbent and J. M. Hammersley ([BH]) and is a branch of probability theory that is especially close to statistical mechanics. Broadbent and Hammersley distinguish between two types of spreading of a fluid through a medium, or between two aspects of the probabilistic models of such processes: diffusion processes, in which the random mechanism is ascribed to the fluid; and percolation processes, in which it is ascribed to the medium. A percolation process typically depends on one or more probabilistic parameters. For example, if molecules of a gas are absorbed at the surface of a porous solid (as in a gas mask) then their ability to penetrate the solid depends on the sizes of the pores in it and their positions, both conceived to be distributed in some random manner. A simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner (that could be determined by a periodic graph), and to be open (thus very large) or closed (thus smaller than the molecules) with probabilities p and 1 − p. As p increases the probability of deeper penetration of the gas into the interior of the solid grows.