  We study, theoretically and numerically, a minimal model for phonons in a
disordered system. For sufficient disorder, the vibrational modes of this
classical system can become Anderson localized, yet this problem has received
significantly less attention than its electronic counterpart. We find rich
behavior in the localization properties of the phonons as a function of the
density, frequency and the spatial dimension. We use a percolation analysis to
argue for a Debye spectrum at low frequencies for dimensions higher than one,
and for a localization/delocalization transition (at a critical frequency)
above two dimensions. We show that in contrast to the behavior in electronic
systems, the transition exists for arbitrarily large disorder, albeit with an
exponentially small critical frequency. The structure of the modes reflects a
divergent percolation length that arises from the disorder in the springs
without being explicitly present in the definition of our model. Within the
percolation approach we calculate the speed-of-sound of the delocalized modes
(phonons), which we corroborate with numerics. We find the critical frequency
of the localization transition at a given density, and find good agreement of
these predictions with numerical results using a recursive Green function
method adapted for this problem. The connection of our results to recent
experiments on amorphous solids are discussed.
