Percolation in a dependent random environment

Draw planes in IR 3 that are orthogonal to the x axis, and intersect the x axis at the points of a Poisson process with intensity ; similarly, draw planes orthogonal to the y and z axes using independent Poisson processes (with the same intensity). Taken together, these planes naturally deene a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of lengthìs open with probability e ?` , and these events are independent given the edge lengths. We show that this model exhibits a phase transition: for large enough there is an innnite open cluster a.s., and for small all open clusters are nite a.s. We prove this result using the method of paths with exponential intersection tails, which is not applicable in two dimensions. The question whether the analogous process in the plane exhibits a phase transition is open.