Seven and a half years of continuous monitoring of giant air showers triggered by ultra high-energy cosmic rays have been recently summarized by the AGASA collaboration. The resulting energy spectrum indicates clearly that the cosmic ray spectrum extends well beyond the Greisen-Zatsepin-Kuzmin (GZK) cut-off at $\sim 5 \times 10^{19}$ eV. Furthermore, despite the small number statistics involved, some structure in the spectrum may be emerging. Using numerical simulations, it is demonstrated in the present work that these features are consistent with a spatial distribution of sources that follows the distribution of luminous matter in the local Universe. Therefore, from this point of view, there is no need for a second high-energy component of cosmic rays dominating the spectrum beyond the GZK cut-off.
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