The contribution of distant sources to the observed flux of ultra
  high-energy cosmic rays
  Ultra-high-energy (UHE) cosmic rays (CRs) interact with cosmic background
radiation through hadronic processes, and the Universe would become `opaque' to
UHE CRs of energies $\sim$($10^{18}$- $10^{20}$) eV over about several tens of
Mpc, setting the Greisen-Zatsepin-Kuz'min (GZK) horizon. We demonstrate that a
non-negligible fraction of the UHE CRs arriving on Earth could originate from
beyond the GZK horizon when heavy nuclear CRs, and the population and evolution
of UHE CR sources are taken into account. We show how the multi-particle CR
horizon is modified by different source populations, and discuss how this leads
to the natural emergence of an isotropic flux component in the observed UHE CR
background. This component would coexist with an anisotropic foreground
component contributed by nearby sources within the GZK horizon.
