  Cosmology models predict that external accretion shocks form in the outer
region of galaxy clusters due to supersonic gas infall from filaments and voids
in the cosmic web. They are characterized by high sonic and Alfv\'enic Mach
numbers, $M_s\sim10-10^2$ and $M_A\sim10^2-10^3$, and propagate into weakly
magnetized plasmas of $\beta\equiv P_g/P_B\gtrsim10^2$. Although strong
accretion shocks are expected to be efficient accelerators of cosmic rays
(CRs), nonthermal signatures of shock-accelerated CRs around clusters have not
been confirmed, and detailed acceleration physics at such shocks has yet to be
understood. In this study, we first establish through two-dimensional
particle-in-cell simulations that at strong high-$\beta$ shocks electrons can
be pre-energized via stochastic Fermi acceleration owing to the ion-Weibel
instability in the shock transition region, possibly followed by injection into
diffusive shock acceleration. Hence, we propose that the models derived from
conventional thermal leakage injection may be employed for the acceleration of
electrons and ions at accretion shocks as well. Applying these analytic models
to numerical shock zones identified in structure formation simulations, we
estimate nonthermal radiation, such as synchrotron and inverse-Compton (IC)
emission due to CR electrons, and $\pi^0$-decay $\gamma$-rays due to CR
protons, around simulated clusters. Our models with the injection parameter,
$Q\approx3.5-3.8$, predict synthetic synchrotron maps, which seem consistent
with recent radio observations of the Coma cluster. However, the detection of
nonthermal IC X-rays and $\gamma$-rays from accretion shocks would be quite
challenging. We suggest that the proposed analytic models may be adopted as
generic recipes for CR production at cosmological shocks.

High Energy Astrophysical Phenomena