  We use simulations of LSS formation to study the build-up of magnetic fields
(MFs) in the ICM. Our basic assumption is that cosmological MFs grow in a MHD
amplification process driven by structure formation out of a seed MF present at
high z. Our LCDM initial conditions for the density fluctuations have been
statistically constrained by the observed galaxies, based on the IRAS 1.2-Jy
all-sky redshift survey. As a result, prominent galaxy clusters in our
simulation coincide closely with their real counterparts. We find excellent
agreement between RMs of our simulated clusters and observational data. The
improved resolution compared to previous work also allows us to study the MF in
large-scale filaments, sheets and voids. By tracing the propagation of UHE
protons in the simulated MF we construct full-sky maps of expected deflection
angles of protons with arrival energies E=1e20eV and 4e19eV, respectively.
Strong deflections are only produced if UHE protons cross clusters, however
covering only a small area on the sky. Multiple crossings of sheets and
filaments over larger distances may give rise to noticeable deflections,
depending on the model adopted for the magnetic seed field. Based on our
results we argue that over a large fraction of the sky the deflections are
likely to remain smaller than the present experimental angular sensitivity.
Therefore, we conclude that forthcoming air shower experiments should be able
to locate sources of UHE protons and shed more light on the nature of
cosmological MFs.
