  We derive analytic results for displacement fields that develop as a response
to external forces on athermal networks in two dimensions. 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 in actively jammed systems. We
show that uncorrelated active forces give rise to long-range correlations in
displacements in athermal systems, with a non-trivial system size dependence.
