  Random, multifield functions can set generic expectations for landscape-style
cosmologies. We consider the inflationary implications of a landscape defined
by a Gaussian random function, which is perhaps the simplest such scenario.
Many key properties of this landscape, including the distribution of saddles as
a function of height in the potential, depend only on its dimensionality, $N$,
and a single parameter, ${\gamma}$, which is set by the power spectrum of the
random function. We show that for saddles with a single downhill direction the
negative mass term grows smaller, relative to the average mass, as $N$
increases, a result with potential implications for the ${\eta}$-problem in
landscape scenarios. For some power spectra Planck-scale saddles have ${\eta}
\sim 1$ and eternal, topological inflation would be common in these scenarios.
Lower-lying saddles typically have large ${\eta}$, but the fraction of these
saddles which would support inflation is computable, allowing us to identify
which scenarios can deliver a universe that resembles ours. Finally, by drawing
inferences about the relative viability of different multiverse proposals we
also illustrate ways in which quantitative analyses of multiverse scenarios are
feasible.
