Statistics of eigenstates near the localization transition on a Random Regular Graph
We study spatially and frequency resolved correlations of wavefunctions (WF) and energy level statistics in Anderson model on the Random Regular Graph (RRG) at criticality and in the delocalized phase. We find a very good agreement between analytical results and numerical approaches, including exact diagonalization and numerical solution of the self-consistency equation for the probability distribution of the local Green functions. We observe correlation length directly in the spatial decay of WF correlations and in the spectral compressibility. Finally, we connect our results to properties of the Anderson model defined on finite dimensional (d) lattices at d >> 1, stressing interesting peculiarities of this limit.