  We consider a random walk on a homogeneous Poisson point process with energy
marks. The jump rates decay exponentially in the A-power of the jump length and
depend on the energy marks via a Boltzmann--like factor. The case A=1
corresponds to the phonon-induced Mott variable range hopping in disordered
solids in the regime of strong Anderson localization. We prove that for almost
every realization of the marked process, the diffusively rescaled random walk,
with arbitrary start point, converges to a Brownian motion whose diffusion
matrix is positive definite, and independent of the environment. Finally, we
extend the above result to other point processes including diluted lattices.
