On the localization transition in symmetric random matrices

We study the behaviour of the inverse participation ratio and the localization transition in infinitely large random matrices through the cavity method. Results are shown for two ensembles of random matrices: Laplacian matrices on sparse random graphs and fully-connected L\'evy matrices. We derive a critical line separating localized from extended states in the case of L\'evy matrices. Comparison between theoretical results and diagonalization of finite random matrices is shown.