  In the first two papers of this series, we characterized the structure of
maximally random jammed (MRJ) sphere packings across length scales by computing
a variety of different correlation functions, spectral functions, hole
probabilities, and local density fluctuations. From the remarkable structural
features of the MRJ packings, especially its disordered hyperuniformity,
exceptional physical properties can be expected. Here, we employ these
structural descriptors to estimate effective transport and electromagnetic
properties via rigorous bounds, exact expansions, and accurate analytical
approximation formulas. These property formulas include interfacial bounds as
well as universal scaling laws for the mean survival time and the fluid
permeability. We also estimate the principal relaxation time associated with
Brownian motion among perfectly absorbing traps. For the propagation of
electromagnetic waves in the long-wavelength limit, we show that a dispersion
of dielectric MRJ spheres within a matrix of another dielectric material forms,
to a very good approximation, a dissipationless disordered and isotropic
two-phase medium for any phase dielectric contrast ratio. We compare the
effective properties of the MRJ sphere packings to those of overlapping
spheres, equilibrium hard-sphere packings, and lattices of hard spheres.
Moreover, we generalize results to micro- and macroscopically anisotropic
packings of spheroids with tensorial effective properties. The analytic bounds
predict the qualitative trend in the physical properties associated with these
structures, which provides guidance to more time-consuming simulations and
experiments. They especially provide impetus for experiments to design
materials with unique bulk properties resulting from hyperuniformity, including
structural-color and color-sensing applications.
