  We compute the distribution of the number of negative eigenvalues (the index)
for an ensemble of Gaussian random matrices, by means of the replica method.
This calculation has important applications in the context of statistical
mechanics of disordered systems, where the second derivative of the potential
energy (the Hessian) is a random matrix whose negative eigenvalues measure the
degree of instability of the energy surface. An analysis of the probability
distribution of the Hessian index is therefore relevant for a geometric
characterization of the energy landscape in disordered systems. The approach we
use here is particularly suitable for this purpose, since it addresses the
problem without any a priori assumption on the random matrix ensemble and can
be naturally extended to more realistic, non-Gaussian distributions.
