Anderson localization in Euclidean random matrices

We study the spectra and localization properties of Euclidean random matrices defined on a random graph. We introduce a powerful method to find the density of states and the localization threshold in topologically disordered (off-lattice) systems. We solve numerically an equation (exact on the random graph) for the probability distribution function of the diagonal elements of the resolvent matrix, with a population dynamics algorithm (PDA). We show that the localization threshold can be estimated by studying the stability of a population of real resolvents under the PDA. An application is given in the context of the instantaneous normal modes of a liquid.