  We calculate the properties of an acoustic polaron in three dimensions in
thermal equilibrium at a given low temperature using the path integral Monte
Carlo method. The specialized numerical method used is described in full
details, thus complementing our previous paper [R. Fantoni, Phys. Rev. B {\bf
86}, 144304 (2012)], and it appears to be the first time it has been used in
this context. Our results are in favor of the presence of a phase transition
from a localized state to an extended state for the electron as the
phonon-electron coupling constant decreases. The phase transition manifests
itself with a jump discontinuity in the potential energy as a function of the
coupling constant and it affects the properties of the path of the electron in
imaginary time: 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.
