Statistics of eigenstates near the localization transition on random regular graphs

Dynamical and spatial correlations of eigenfunctions as well as energy level correlations in the Anderson model on random regular graphs (RRG) are studied. We consider the critical point of the Anderson transition and the delocalized phase. In the delocalized phase near the transition point, the observables show a broad critical regime for system sizes $N$ below the correlation volume $N_{\xi}$ and then cross over to the ergodic behavior. Eigenstate correlations allow us to visualize the correlation length $\xi \sim \log N_{\xi}$ that controls the finite-size scaling near the transition. The critical-to-ergodic crossover is very peculiar, since the critical point is similar to the localized phase, whereas the ergodic regime is characterized by very fast "diffusion" which is similar to the ballistic transport. In particular, the return probability crosses over from a logarithmically slow variation with time in the critical regime to an exponentially fast decay in the ergodic regime. Spectral correlations in the delocalized phase near the transition are characterized by level number variance $\Sigma_2(\omega)$ crossing over, with increasing freqyency $\omega$, from ergodic behavior $\Sigma_2=\left(2/\pi^2\right)\ln\omega/\Delta$ to $\Sigma_2\propto \omega^2$ at $\omega_c\sim (N N_{\xi})^{-1/2}$ and finally to Poissonian behavior $\Sigma_2 = \omega/\Delta$ at $\omega_{\xi}\sim N_{\xi}^{-1}$. We find a perfect agreement between results of exact diagonalization and those resulting from the solution of the self-consistency equation obtained within the saddle-point analysis of the effective supersymmetric action. We show that the RRG model can be viewed as an intricate $d\to\infty$ limit of the Anderson model in $d$ spatial dimensions.