Propagation of Ultra-High Energy Cosmic Rays above $10^{19}$ eV in a   Structured Extragalactic Magnetic Field and Galactic Magnetic Field
  We present numerical simulations on propagation of Ultra-High Energy Cosmic Rays (UHECRs) above $10^{19}$ eV in a structured extragalactic magnetic field (EGMF) and simulate their arrival distributions at the earth. We use the IRAS PSCz catalogue in order to construct a model of the EGMF and source models of UHECRs, both of which reproduce the local structures observed around the Milky Way. We also consider modifications of UHECR arrival directions by the galactic magnetic field. We follow an inverse process of their propagation from the earth and record the trajectories. This enables us to calculate only trajectories of UHECRs arriving at the earth, which saves the CPU time. From these trajectories and our source models, we construct arrival distributions of UHECRs and calculate the harmonic amplitudes and the two point correlation functions of them. We estimate number density of sources which reproduces the Akeno Ground Air Shower Array (AGASA) observation best. As a result, we find that the most appropriate number density of the sources is $\sim 5 \times 10^{-6}$ Mpc$^{-3}$. This constrains the source candidates of UHECRs. We also demonstrate skymaps of their arrival distribution with the event number expected by future experiments and examine how the EGMF affects their arrival distribution. A main result is diffusion of clustering events which are obtained from calculations in the absence of the EGMF. This tendency allows us to reproduce the observed two point correlation function better. 
Hajime Takami, Hiroyuki Yoshiguchi and Katsuhiko Sato