The wasteland of random supergravities

We show that in a general N = 1 supergravity with N >> 1 scalar fields, an exponentially small fraction of the de Sitter critical points are metastable vacua. Taking the superpotential and Kahler potential to be random functions, we construct a random matrix model for the Hessian matrix, which is well-approximated by the sum of a Wigner matrix and two Wishart matrices. We compute the eigenvalue spectrum analytically from the free convolution of the constituent spectra and find that in typical configurations, a significant fraction of the eigenvalues are negative. Building on the Tracy-Widom law governing fluctuations of extreme eigenvalues, we determine the probability P of a large fluctuation in which all the eigenvalues become positive. Strong eigenvalue repulsion makes this extremely unlikely: we find P proportional to exp(-c N-p), with c, p being constants. For generic critical points we find p approximate to 1.5, while for approximately-supersymmetric critical points, p approximate to 1.3. Our results have significant implications for the counting of de Sitter vacua in string theory, but the number of vacua remains vast.