  Besides the dynamical slowing down signaled by an enormous increase of the
viscosity approaching the glass transition, structural glasses show interesting
anomalous thermodynamic features at low temperatures that hint at peculiar
deviations from Debye's law at low enough frequencies. Theory, numerical
simulations, and experiments suggest that deviation from Debye's law is due to
soft-localized glassy modes that populate the low-frequency spectrum. We study
the localization properties of the low-frequency modes in a three-dimensional
supercooled liquid model. The density of states $D(\omega)$ is computed
considering the inherent structures of configurations well thermalized at
parental temperatures close to the dynamical transition $T_\text d$. We observe
a crossover in the probability distribution of the inverse of the participation
ratio that happens approaching $T_\text d$ from high temperatures. We show that
a similar crossover is observed at high parental temperature when the
translational invariance of the system is explicitly broken by a random pinning
field.

Disordered Systems and Neural Networks;Soft Condensed Matter;Statistical Mechanics