  We discuss the order of the variance on a lattice analogue of the Hammersley
process with boundaries, for which the environment on each site has
independent, Bernoulli distributed values. The last passage time is the maximum
number of Bernoulli points that can be collected on a piecewise linear path,
where each segment has strictly positive but finite slope. We show that along
characteristic directions the order of the variance of the last passage time is
of order $N^{2/3}$ in the model with boundary. These characteristic directions
are restricted in a cone starting at the origin, and along any direction
outside the cone, the order of the variance changes to $O(N)$ in the boundary
model and to $O(1)$ for the non-boundary model. This behaviour is the result of
the two flat edges of the shape function.
