  This is the second, and last paper in which we address the behavior of
oriented first passage percolation on the hypercube in the limit of large
dimensions. We prove here that the extremal process converges to a Cox process
with exponential intensity. This entails, in particular, that the first passage
time converges weakly to a random shift of the Gumbel distribution. The random
shift, which has an explicit, universal distribution related to modified Bessel
functions of the second kind, is the sole manifestation of correlations ensuing
from the geometry of Euclidean space in infinite dimensions. The proof combines
the multiscale refinement of the second moment method with a conditional
version of the Chen-Stein bounds, and a contraction principle.
