Math 778 — Spring 2002 Stochastic Processes

Imagine that the edges of the two-dimensional (n-dimensional) lattice are the inner passageways of a large porous stone. Some passages are open (broad enough for oil to pass along them), and some are not. For lack of better knowledge, assume that each edge is open with probability p and closed otherwise, independently of all other edges. Let the origin represent the center of the stone, and immerse the stone in oil. Then the event that oil reaches the center of the stone corresponds to the event that there exists a path of open edges connecting the center to the boundary, that is, the origin to the infinity. What is the chance of the above event? How does it depend on p? This course will study these and related questions. Percolation theory is a great source of easy-to-state problems with beautiful, illuminating and often challenging solutions.

Text:Percolation by G. Grimmett, 2nd edition, Springer, 1999.