  We calculate the low temperature properties of an acoustic polaron in three
dimensions in thermal equilibrium at a given temperature using a specialized
path integral Monte Carlo method. In particular we find numerical evidence that
the chosen Hamiltonian for the acoustic polaron describes a phase transition
from a localized state to an unlocalized state for the electron as the
phonons-electron coupling constant decreases. The phase transition manifests
itself with a jump discontinuity in the potential energy as a function of the
coupling constant. In the weak coupling regime the electron is in an extended
state whereas in the strong coupling regime it is found in a self-trapped
state.
