  We introduce and study a new class of Banded Random Matrix model describing
sparse, long range quantum hopping in one dimension. Using a series of analytic
arguments, numerical simulations, and mappings to statistical physics models,
we establish the phase diagram of the model. A genuine localisation transition,
with well defined mobility edges, appears as the hopping rate decreases slower
than $\ell^{-2}$, where $\ell$ is the distance. Correspondingly, the decay of
the localised states evolves from a standard exponential shape to a stretched
exponential and finally to a novel $\exp(-C\ln^\kappa \ell)$ behaviour, with
$\kappa > 1$.
