(300c) Applications of Conformal and Number Theory to Problems in Percolation Crossing Probabilities

Percolation relates to a variety of problems in chemical engineering. The question of the crossing probability at criticality has been solved in 1992 by Cardy for a rectangular system, and by conformal transformation it can be solved for a system of arbitrary shape. We consider a parallelogram, and find that Ramanujan's theory of elliptic functions to alternate bases plays a central role. We also look at the question of the density of the percolated region anchored to a point on the boundary (say, the expected intrusion of fluid from a single point) and compare numerical simulations with theory. For the density of clusters anchored to multiple points, the solution turns out to be the square root of the product of densities from individual points. This is demonstrated by beautiful computer-generated illustrations.