  Particulate physical gels are sparse, low-density amorphous materials in
which clusters of glasses are connected to form a heterogeneous network
structure. This structure is characterized by two length scales, $\xi_s$ and
$\xi_G$: $\xi_s$ measures the length of heterogeneities in the network
structure, and $\xi_G$ is the size of glassy clusters. Accordingly, the
vibrational states of such a material also exhibit a multiscale nature with two
characteristic frequencies, $\omega_\ast$ and $\omega_G$, which are associated
with $\xi_s$ and $\xi_G$, respectively: (i) phonon-like vibrations in the
homogeneous medium at $\omega < \omega_\ast$, (ii) phonon-like vibrations in
the heterogeneous medium at $\omega_\ast < \omega < \omega_G$, and (iii)
disordered vibrations in the glassy clusters at $\omega > \omega_G$. Here, we
demonstrate that the multiscale characteristics seen in the static structures
and vibrational states also extend to the phonon transport properties. Phonon
transport exhibits two distinct crossovers at the frequencies $\omega_\ast$ and
$\omega_G$~(or at wavenumbers of $\sim \xi_s^{-1}$ and $\sim \xi_G^{-1}$). In
particular, both transverse and longitudinal phonons cross over between
Rayleigh scattering at $\omega < \omega_\ast$ and diffusive damping at
$\omega>\omega_\ast$. Remarkably, the Ioffe--Regel limit is located at the very
low frequency of $\omega_\ast$. Thus, phonon transport is localized above
$\omega_\ast$, even where phonon-like vibrational states persist. This markedly
strong scattering behavior is caused by the sparse, porous structure of the
gel.
