  We consider the statistical properties of vacua and inflationary trajectories
associated with a random multifield potential. Our underlying motivation is the
string landscape, but our calculations apply to general potentials. Using
random matrix theory, we analyze the Hessian matrices associated with the
extrema of this potential. These potentials generically have a vast number of
extrema. If the cross-couplings (off-diagonal terms) are of the same order as
the self-couplings (diagonal terms) we show that essentially all extrema are
saddles, and the number of minima is effectively zero. Avoiding this requires
the same separation of scales needed to ensure that Newton's constant is stable
against radiative corrections in a string landscape. Using the central limit
theorem we find that even if the number of extrema is enormous, the typical
distance between extrema is still substantial -- with challenging implications
for inflationary models that depend on the existence of a complicated path
inside the landscape.
