Number of distinct sites visited by Lévy flights injected into a $d$-dimensional lattice

We study the average number of distinct sites $〈S_{N_0}\mathrm{}\left(t\right)\mathrm{}〉$ visited by Lévy flights injected in the center of a lattice: $N_0$ new particles appear in the center of the lattice at each time step. Lévy flights are particles which have the probability $p\left(\mathcal{l}\right)=A{\mathcal{l}}^{-(1+\alpha )},0<\alpha <2\mathrm{}$ of making an $\mathcal{l}$-length jump. We show analytically that the asymptotic form of $〈S_{N_0}\mathrm{}\left(t\right)\mathrm{}〉$ is related to that of the case of constant initial number $N$ of particles. We find that different ranges of $\alpha$ correspond to different limits, $\overrightarrow{t}\infty$ and $\overrightarrow{N}\infty$, in the behavior of the number of sites visited by constant number of particles. The results obtained analytically are in good agreement with Monte Carlo simulations. We also discuss possible results for $\alpha >~2$.