  The transport of excitations between pinned particles in many physical
systems may be mapped to single-particle models with power-law hopping,
$1/r^a$. For randomly spaced particles, these models present an effective
peculiar disorder that leads to surprising localization properties. We show
that in one-dimensional systems almost all eigenstates (except for a few states
close to the ground state) are power-law localized for any value of $a>0$.
Moreover, we show that our model is an example of a new universality class of
models with power-law hopping, characterized by a duality between systems with
long-range hops ($a<1$) and short-range hops ($a>1$) in which the wave function
amplitude falls off algebraically with the same power $\gamma$ from the
localization center.
