  We consider the random walk on a simple point process on $\Bbb{R}^d$,
$d\geq2$, whose jump rates decay exponentially in the $\alpha$-power of jump
length. The case $\alpha =1$ 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 $\alpha\in(0,d)$,
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 $L^2$. For the
Poisson point process, we prove that at $\alpha=d$, there is a transition from
diffusive to subdiffusive behavior of the mixing time.
