  The superconducting (SC) and charge-density-wave (CDW) susceptibilities of
the two dimensional Holstein model are computed using determinant quantum Monte
Carlo (DQMC), and compared with results computed using the Migdal-Eliashberg
(ME) approach. We access temperatures as low as 25 times less than the Fermi
energy, $E_F$, which are still above the SC transition. We find that the SC
susceptibility at low $T$ agrees quantitatively with the ME theory up to a
dimensionless electron-phonon coupling $\lambda_0 \approx 0.4$ but deviates
dramatically for larger $\lambda_0$. We find that for large $\lambda_0$ and
small phonon frequency $\omega_0 \ll E_F$ CDW ordering is favored and the
preferred CDW ordering vector is uncorrelated with any obvious feature of the
Fermi surface.
