  We study the response of a two-dimensional hexagonal packing of massless,
rigid, frictionless spherical grains due to a vertically downward point force
on a single grain at the top layer. We use a statistical approach, where each
mechanically stable configuration of contact forces is equally likely. We show
that this problem is equivalent to a correlated $q$-model. We find that the
response is double-peaked, where the two peaks, sharp and single-grain diameter
wide, lie on the two downward lattice directions emanating from the point of
the application of the external force. For systems of finite size, the
magnitude of these peaks decreases towards the bottom of the packing, while
progressively a broader, central maximum appears between the peaks. The
response behaviour displays a remarkable scaling behaviour with system size
$N$: while the response in the bulk of the packing scales as $\frac{1}{N}$, on
the boundary it is independent of $N$, so that in the thermodynamic limit only
the peaks on the lattice directions persist. This qualitative behaviour is
extremely robust, as demonstrated by our simulation results with different
boundary conditions. We have obtained expressions of the response and higher
correlations for any system size in terms of integers corresponding to an
underlying discrete structure.

Statistical Mechanics