Percolation in a dependent random environment

Draw planes in R-3 that are orthogonal to the x axis, and intersect the x a xis at the points of a Poisson process with intensity lambda; similarly, dr aw planes orthogonal to the y and z axes using independent Poisson processe s (with the same intensity). Taken together, these planes naturally define a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of length l is open with probability e(-l), and th ese events are independent given the edge lengths. We show that this model exhibits a phase transition: for large enough lambda there is an infinite o pen cluster a.s., and for small lambda all open clusters are finite a.s. We prove this result using the method of paths with exponential intersection rails, which is not applicable in two dimensions. The question whether the analogous process in the plane exhibits a phase transition is open. (C) 200 0 John Wiley & Sons, Inc.