  The heterogeneous force networks in static granular media --- formed from
contact forces between grains and spanning from boundary to boundary in the
packing --- are distinguished from other network structures in that they must
satisfy constraints of mechanical equilibrium on every vertex/grain. Here we
study the statistics of ensembles of hyperstatic frictionless force networks,
which are composed of more forces than can be determined uniquely from force
balance. Hyperstatic force networks possess degrees of freedom that rearrange
one balanced network into another. We construct these rearrangements, count
them, identify their elementary building blocks, and show that in two
dimensions they are related via duality to so-called floppy modes, which play
an important role in many other aspects of granular physics. We demonstrate
that the number of rearrangements governs the macroscopic statistical
properties of the ensemble, in particular the macroscopic fluctuations of
stress, which scale with distance to the isostatic point. We then show that a
maximum entropy postulate allows one to quantitatively capture many features of
the microscopic statistics. Boundaries are shown to influence the statistics
strongly: the probability distribution of large forces can have a qualitatively
different form on the boundary and in the bulk. Finally, we consider the role
of spatial correlations and dimension. All predictions are tested against
highly accurate numerical simulations of the ensemble, performed using umbrella
sampling.
