The Distribution of Vacua in Random Landscape Potentials
  Landscape cosmology posits the existence of a convoluted, multidimensional,
scalar potential -- the "landscape" -- with vast numbers of metastable minima.
Random matrices and random functions in many dimensions provide toy models of
the landscape, allowing the exploration of conceptual issues associated with
these scenarios. We compute the relative number and slopes of minima as a
function of the vacuum energy $\Lambda$ in an $N$-dimensional Gaussian random
potential, quantifying the associated probability density, $p(\Lambda)$. After
normalisations $p(\Lambda)$ depends only on the dimensionality $N$ and a single
free parameter $\gamma$, which is related to the power spectrum of the random
function. For a Gaussian landscape with a Gaussian power spectrum, the fraction
of positive minima shrinks super-exponentially with $N$; at $N=100$,
$p(\Lambda>0) \approx 10^{-1197}$. Likewise, typical eigenvalues of the Hessian
matrices reveal that the flattest approaches to typical minima grow flatter
with $N$, while the ratio of the slopes of the two flattest directions grows
with $N$. We discuss the implications of these results for both swampland and
conventional anthropic constraints on landscape cosmologies. In particular, for
parameter values when positive minima are extremely rare, the flattest
approaches to minima where $\Lambda \approx 0$ are much flatter than for
typical minima, increasingly the viability of quintessence solutions.
