ABSENCE OF GEODESICS IN FIRST-PASSAGE PERCOLATION ON A HALF-PLANE

An H-geodesic is a doubly infinite path which locally minimizes the pa ssage time in the i.i.d. first passage percolation model on a half-pla ne H. Under the assumption that the bond passage times are continuousl y 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 Z(2) has to intersect all straight lines with rational slopes.