  The orthant model is a directed percolation model on $\mathbb{Z}^d$, in which
all clusters are infinite. We prove a sharp threshold result for this model: if
$p$ is larger than the critical value above which the cluster of $0$ is
contained in a cone, then the shift from $0$ that is required to contain the
cluster of $0$ in that cone is exponentially small. As a consequence, above
this critical threshold, a shape theorem holds for the cluster of $0$, as well
as ballisiticity of the random walk on this cluster.
