Force distributions and stress fluctuations in a triangular lattice of rigid bars

We study the uniformly weighted ensemble of force balanced configurations on a triangular network of nontensile contact forces as a model of force distribution on a hyperstatic granular material. For periodic boundary conditions corresponding to isotropic compressive stress, the probability distribution for single-contact forces, $P(f)$, decays faster than exponentially, and a field closely related to the lattice version of the Airy stress function is found to have fluctuations characterized by a structure factor $S(q) \sim 1/q^4$. The super-exponential decay of $P(f)$ persists in lattices diluted to the rigidity percolation threshold. On the other hand, for anisotropic imposed stresses, a broader tail emerges, becoming a pure exponential in the limit of infinite lattice size and infinitely strong anisotropy.