  We give an asymptotic evaluation of the complexity of spherical p-spin
spin-glass models via random matrix theory. This study enables us to obtain
detailed information about the bottom of the energy landscape, including the
absolute minimum (the ground state), the other local minima, and describe an
interesting layered structure of the low critical values for the Hamiltonians
of these models. We also show that our approach allows us to compute the
related TAP-complexity and extend the results known in the physics literature.
As an independent tool, we prove a LDP for the k-th largest eigenvalue of the
GOE, extending the results of Ben Arous, Dembo and Guionnett (2001).
