Existence and coalescence of directed infinite geodesics in the percolation cone for Durrett-Liggett class of measures

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.