Ken Alexander: Disjointness of geodesics for first passage percolation in the plane (USC)

First passage percolation may be viewed as a way of creating a random metric on the integer lattice. Random passage times (iid) are assigned to the edges of the lattice, and the distance from x to y is defined to be the smallest sum of passage times among all paths from x to y. We consider geodesics (shortest paths) in this context; specifically, what is the probability that two close--by nearly--parallel geodesics are disjoint? More precisely, if the geodesics have end-to-end distance L and the endpoints are separated by distance a at one end and b at the other, how does the disjointness probability vary with L,a,b? We answer this question, at least at the level of the proper exponents for L,a, and b.