Isoperimetric inequalities and mixing time for a random walk on a random point process

We consider the random walk on a simple point process on .d, d.2, whose jump rates decay exponentially in the .-power of jump length. The case .=1 corresponds to the phonon-induced variable-range hopping in disordered solids in the regime of strong Anderson localization. Under mild assumptions on the point process, we show, for ..(0, d), that the random walk confined to a cubic box of side L has a.s. Cheeger constant of order at least L.1 and mixing time of order L2. For the Poisson point process, we prove that at .=d, there is a transition from diffusive to subdiffusive behavior of the mixing time.