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 $N=100$, $p(\Lambda>0) \approx 10^{-780}$. 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.