Supersymmetric vacua in random supergravity

We determine the spectrum of scalar masses in a supersymmetric vacuum of a general N = 1 supergravity theory, with the Kahler potential and superpotential taken to be random functions of N complex scalar fields. We derive a random matrix model for the Hessian matrix and compute the eigenvalue spectrum. Tachyons consistent with the Breitenlohner-Freedman bound are generically present, and although these tachyons cannot destabilize the supersymmetric vacuum, they do influence the likelihood of the existence of an 'uplift' to a metastable vacuum with positive cosmological constant. We show that the probability that a supersymmetric AdS vacuum has no tachyons is formally equivalent to the probability of a large fluctuation of the smallest eigenvalue of a certain real Wishart matrix. For normally-distributed matrix entries and any N, this probability is given exactly by P = exp(-2N(2)vertical bar W vertical bar(2)/m(susy)(2)), with W denoting the superpotential and m(susy) the supersymmetric mass scale; for more general distributions of the entries, our result is accurate when N >> 1. We conclude that for vertical bar W vertical bar greater than or similar to m(susy)/N, tachyonic instabilities are ubiquitous in configurations obtained by uplifting supersymmetric vacua.