  In Poisson percolation each edge becomes open after an independent
exponentially distributed time with rate that decreases in the distance from
the origin. As a sequel to our work on the square lattice, we describe the
limiting shape of the component containing the origin in the oriented case. We
show that the density of occupied sites at height $y$ in the cluster is close
to the percolation probability in the corresponding homogeneous percolation
process, and we study the fluctuations of the boundary.
