Index distribution of random matrices with an application to disordered systems

We compute the distribution of the number of negative eigenvalues (the inde x) for an ensemble of Gaussian random matrices, by means of the replica met hod. This calculation has important applications in the context of statisti cal mechanics of disordered systems, where the second derivative of the pot ential energy (the Hessian) is a random matrix whose negative eigenvalues m easure the degree of instability of the energy surface. An analysis of the probability distribution of the Hessian index is therefore relevant for a g eometric characterization of the energy landscape in disordered systems. Th e 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 dist ributions.