  For first passage percolation (FPP) on integer lattice with i.i.d. passage
time distributions, in order to show existence of semi-infinite geodesics along
a fixed direction, one requires unproven assumptions on the limiting shape. We
consider FPP on two-dimensional integer lattice with i.i.d. passage times
distributed as Durrett-Liggett class of measures. For this model, we show that
along any direction in a deterministic angular sector (known as percolation
cone), starting from every lattice point there exists an infinite geodesic
along that direction and such directed geodesics coalesce almost surely. We
prove that for this model, bi-infinite geodesics exist almost surely. Our proof
does not require any assumption on the limiting shape.
