Quantum chaos, localization and phase transitions in random graphs

The energy level statistics of uniform random graphs are studied, by treating the graphs as random tight-binding lattices. The inherent random geometry of the graphs and their dynamical spatial dimensionality, leads to various quantum chaotic and localized phases and transitions between them. Essentially the random geometry acts as disorder, whose strength is characterized by the ratio of edges over vertices R in the graphs. For dense graphs, with large ratio R, the spacing between successive energy levels follows the Wigner-Dyson distribution, leading to a quantum chaotic behavior and a metallic phase, characterized by level repulsion. For ratios near R=0.5, where a large dominating component in the graph appears, the level spacing follows the Poisson distribution with level crossings and a localized phase for the respective wavefunctions lying on the graph. For intermediate ratios R we observe a phase transition between the quantum chaotic and localized phases characterized by a semi-Poisson distribution. The values R of the critical regime where the phase transition occurs depend on the energy of the system. Our analysis shows that physical systems with random geometry, for example ones with a fluctuating/dynamical spatial dimension, contain novel universal phase transition properties, similar to those occuring in more traditional phase transitions based on symmetry breaking mechanisms, whose universal properties are strongly determined by the dimensionality of the system.