  We derive exact results for displacement fields that develop as a response to
external pinning forces in two dimensional athermal networks. For a triangular
lattice arrangement of particles interacting through soft potentials, we
develop a Green's function formalism which we use to derive exact results for
displacement fields produced by localized external forces. We show that in the
continuum limit the displacement fields decay as $1/r$ at large distances $r$
away from a force dipole. Finally, we extend our formulation to study
correlations in the displacement fields produced by the external pinning
forces. We show that uncorrelated pinned forces at each vertex give rise to
long-range correlations in displacements in athermal systems, with a
non-trivial system size dependence. We verify our predictions with numerical
simulations of athermal networks in two dimensions.
