  We analyze the propagation of excitons in a $d$-dimensional lattice with
power-law hopping $\propto 1/r^\alpha$ in the presence of dephasing, described
by a generalized Haken-Strobl-Reineker model. We show that in the strong
dephasing (quantum Zeno) regime the dynamics is described by a classical master
equation for an exclusion process with long jumps. In this limit, we
analytically compute the spatial distribution, whose shape changes at a
critical value of the decay exponent $\alpha_{\rm cr} = (d+2)/2$. The exciton
always diffuses anomalously: a superdiffusive motion is associated to a L\'evy
stable distribution with long-range algebraic tails for $\alpha\leq\alpha_{\rm
cr}$, while for $\alpha > \alpha_{\rm cr}$ the distribution corresponds to a
surprising mixed Gaussian profile with long-range algebraic tails, leading to
the coexistence of short-range diffusion and long-range L\'evy-flights. In the
many-exciton case, we demonstrate that, starting from a domain-wall exciton
profile, algebraic tails appear in the distributions for any $\alpha$, which
affects thermalization: the longer the hopping range, the faster equilibrium is
reached. Our results are directly relevant to experiments with cold trapped
ions, Rydberg atoms and supramolecular dye aggregates. They provide a way to
realize an exclusion process with long jumps experimentally.
