  We analyze the fluctuations in particle positions and inter-particle forces
in disordered jammed crystals in the limit of weak disorder. We demonstrate
that such athermal systems are fundamentally different from their thermal
counterparts, characterized by constrained fluctuations of forces perpendicular
to the lattice directions. We develop a disorder perturbation expansion in
polydispersity about the crystalline state, which we use to derive exact
results to linear order. We show that constrained fluctuations result as a
consequence of local force balance conditions, and are characterized by
non-Gaussian distributions which we derive exactly. We analytically predict
several properties of such systems, including the scaling of the average
coordination with polydispersity and packing fraction, which we verify with
numerical simulations using soft disks with one-sided harmonic interactions.
