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

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We consider the random walk on a simple point process on Rᵈ, d≥2, whose jump rates decay exponentially in the α-power of jump length. The case α=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 α∈(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 α=d, there is a transition from diffusive to subdiffusive behavior of the mixing time.