The GZK Bound in Discrete Space

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The maximum distance bound for ultrahigh energy cosmic rays (UHECR) with energies above the Greisen-Zatsepin-Kuzmich cutoff $\sim 10^{19} eV$ relaxes significantly if there is some mechanism that forces UHECR to propagate in discrete intervals, rather than continuously, through intergalactic space. In particular, intervals as small as a femtometer relax the bound for protons by an order of magnitude and potentially account for the observed excess of UHECR flux.
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