The distribution of vacua in random landscape potentials

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