  The half-orthant model is a partially oriented model of a random medium
involving a parameter $p\in [0,1]$, for which there is a critical value
$p_c(d)$ (depending on the dimension $d$) below which every point is reachable
from the origin. We prove a limit theorem for the graph-distance (or "chemical
distance") for this model when $p<p_c(2)$, and also when $1-p$ is larger than
the critical parameter for site percolation in $\mathbb{Z}^d$. The proof
involves an application of the subadditive ergodic theorem. Novel arguments
herein include the method of proving that the expected number of steps to reach
any given point is finite, as well as an argument that is used to show that the
shape is "non-trivial" in certain directions.

Probability