  We show that in a general \cal{N} = 1 supergravity with N \gg 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 \propto exp(-c
N^p), with c, p being constants. For generic critical points we find p \approx
1.5, while for approximately-supersymmetric critical points, p \approx 1.3. Our
results have significant implications for the counting of de Sitter vacua in
string theory, but the number of vacua remains vast.
