Localization of eigenvectors on random graphs

Abstract: The work is motivated by experiments on localization of classical waves. i. e. light and sound, in random media. Especially we have in mind granular materials, where waves travel along complex structures of force chains. These force chains are modelled by random graphs of various types. We investigate first the most classical Erdős-Rényi graph ensemble and then compare it to random regular (especially cubic) graphs and to scale-free bipartite graphs. The methodology is based on exact diagonalization of large samples. However, for ER graphs, we present also analytical results using the cavity method and effective medium approximation (analogy of CPA for the case of off-diagonal disorder). We clearly demonstrate the presence of localization due to purely topological disorder. The mobility edge is located in the Lifschitz tail beyond the effective medium (or CPA) edge. The model of random bipartite graph is interpreted in the language of jamming transition in a granular material. We show how the two transitions, namely fluid-jammed and localized-delocalized, compete with each other. We also touch some open problems related to the phenomenon of many-body localization, where the complex topology of Fock space is approximated by a random graph. Finally, we show some surprising interdisciplinary applications.