Absence of geodesics in first-passage percolation on a half-plane

Abstract

An H-geodesic is a doubly infinite path which locally minimizes the passage time in the i.i.d. first passage percolation model on a half-plane H. Under the assumption that the bond passage times are continuously distributed with a finite mean, we prove that, with probability 1, H-geodesics do not exist. As a corollary we show that, with probability 1, any geodesic in the analogous model on the whole plane $\mathbf{Z}^2$ has to intersect all straight lines with rational slopes.