Georgia Institute of Technology College of Sciences

Several well-known problems in first-passage percolation relate to the behavior of infinite geodesics: whether they coalesce and how rapidly, and whether doubly infinite "bigeodesics'' exist. In the plane, a version of coalescence of "parallel'' geodesics has previously been shown; we will discuss new results that show infinite geodesics from the origin have zero density in the plane. We will describe related forthcoming work showing that geodesics coalesce in dimensions three and higher, under unproven assumptions believed to hold below the model's upper critical dimension. If time permits, we will also discuss results on the bigeodesic question in dimension three and higher.