  We study a particular model of a random medium, called the orthant model, in
general dimensions $d\ge 2$. Each site $x\in \Z^d$ independently has arrows
pointing to its positive neighbours $x+e_i$, $i=1,\dots, d$ with probability
$p$ and otherwise to its negative neighbours $x-e_i$, $i=1,\dots, d$ (with
probability $1-p$). We prove a shape theorem for the set of sites reachable by
following arrows, starting from the origin, when $p$ is large. The argument
uses subadditivity, as would be expected from the shape theorems arising in the
study of first passage percolation. The main difficulty to overcome is that the
primary objects of study are not stationary, which is a key requirement of the
subadditive ergodic theorem.
