We present numerical simulations on the propagation of UHE protons with energies of ( 10 ^ { 19.5 } -10 ^ { 22 } ) eV in extragalactic magnetic fields over 1 Gpc .
We use the ORS galaxy sample , which allow us to accurately quantify the contribution of nearby sources to the energy spectrum and the arrival distribution , as a source model .
The sample is corrected taking the selection effect and absence of galaxies in the zone of avoidance ( |b| < 20 ^ { \circ } ) into account .
We calculate three observable quantities , cosmic ray spectrum , harmonic amplitude , and two point correlation function from our data of numerical simulations .
With these quantities , we compare the results of our numerical calculations with the observation .
We find that the arrival distribution of UHECRs become to be most isotropic as restricting sources to luminous galaxies ( M _ { lim } = -20.5 ) .
However , it is not isotropic enough to be consistent with the AGASA observation , even for M _ { lim } = -20.5 .
In order to obtain sufficiently isotropic arrival distribution , we randomly select sources , which contribute to the observed cosmic ray flux , from the ORS sample more luminous than -20.5 mag , and investigate dependence of the results on their number .
We show that the three observable quantities including the GZK cutoff of the energy spectrum can be reproduced in the case that the number fraction \sim 10 ^ { -1.7 } of the ORS galaxies more luminous than -20.5 mag is selected as UHECR sources .
In terms of the source number density , this constraint corresponds to \sim 10 ^ { -6 } Mpc ^ { -3 } .
However , since mean number of sources within the GZK sphere is only \sim 0.5 in this case , the AGASA 8 events above 10 ^ { 20.0 } eV , which do not constitute the clustered events with each other , can not be reproduced .
On the other hand , if the cosmic ray flux measured by the HiRes , which is consistent with the GZK cutoff , is correct and observational features about the arrival distribution of UHECRs are same as the AGASA , our source model can explain both the arrival distribution and the flux at the same time .
Thus , we conclude that large fraction of the AGASA 8 events above 10 ^ { 20 } eV might originate in the topdown scenarios , or that the cosmic ray flux measured by the HiRes experiment might be better .
We also discuss the origin of UHECRs below 10 ^ { 20.0 } eV through comparisons between the number density of astrophysical source candidates and our result ( \sim 10 ^ { -6 } Mpc ^ { -3 } ) .