  Under decompression, disordered solids undergo an unjamming transition where
they become under-coordinated and lose their structural rigidity. The
mechanical and vibrational properties of these materials have been an object of
theoretical, numerical, and experimental research for decades. In the study of
low-coordination solids, understanding the behavior and physical interpretation
of observables that diverge near the transition is of particular importance.
Several such quantities are length scales ($\xi$ or $l$) that characterize the
size of excitations, the decay of spatial correlations, the response to
perturbations, or the effect of physical constraints in the boundary or bulk of
the material. Additionally, the spatial and sample-to-sample fluctuations of
macroscopic observables such as contact statistics or elastic moduli diverge
approaching unjamming. Here, we discuss important connections between all of
these quantities, and present numerical results that characterize the scaling
properties of sample-to-sample contact and shear modulus fluctuations in
ensembles of low-coordination disordered sphere packings and spring networks.
Overall, we highlight three distinct scaling regimes and two crossovers in the
disorder quantifiers $\chi_z$ and $\chi_\mu$ as functions of system size $N$
and proximity to unjamming $\delta z$. As we discuss, $\chi_X$ relates to the
standard deviation $\sigma_X$ of the sample-to-sample distribution of the
quantity $X$ (e.g. excess coordination $\delta z$ or shear modulus $\mu$) for
an ensemble of systems. Importantly, $\chi_\mu$ has been linked to
experimentally accessible quantities that pertain to sound attenuation and the
density of vibrational states in glasses.
