  We study the response of a two-dimensional hexagonal packing of 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
configuration of the contact forces is equally likely. We show that this
problem is equivalent to a correlated $q$-model. We find that the response
displays two peaks which lie precisely on the downward lattice directions
emanating from the point of application of the force. With increasing depth,
the magnitude of the peaks decreases, and a central peak develops. On the
bottom of the pile, only the middle peak persists. The response of different
system sizes exhibits self-similarity.
