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We analyze the propagation of excitons in a $d$-dimensional lattice withpower-law hopping $\propto 1/r^\alpha$ in the presence of dephasing, describedby a generalized Haken-Strobl-Reineker model. We show that in the strongdephasing (quantum Zeno) regime the dynamics is described by a classical masterequation for an exclusion process with long jumps. In this limit, weanalytically compute the spatial distribution, whose shape changes at acritical value of the decay exponent $\alpha_{\rm cr} = (d+2)/2$. The excitonalways diffuses anomalously: a superdiffusive motion is associated to a L\'evystable distribution with long-range algebraic tails for $\alpha\leq\alpha_{\rmcr}$, while for $\alpha > \alpha_{\rm cr}$ the distribution corresponds to asurprising mixed Gaussian profile with long-range algebraic tails, leading tothe coexistence of short-range diffusion and long-range L\'evy-flights. In themany-exciton case, we demonstrate that, starting from a domain-wall excitonprofile, algebraic tails appear in the distributions for any $\alpha$, whichaffects thermalization: the longer the hopping range, the faster equilibrium isreached. Our results are directly relevant to experiments with cold trappedions, Rydberg atoms and supramolecular dye aggregates. They provide a way torealize an exclusion process with long jumps experimentally.

Anomalous diffusion in the Long-Range Haken-Strobl-Reineker model