  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.
