Level compressibility for the Anderson model on regular random graphs and the eigenvalue statistics in the extended phase

We calculate the level compressibility χ(W,L) of the energy levels inside [−L/2,L/2] for the Anderson model on infinitely large random regular graphs with on-site potentials distributed uniformly in [−W/2,W/2]. We show that χ(W,L) approaches the limit limL→0+χ(W,L)=0 for a broad interval of the disorder strength W within the extended phase, including the region of W close to the critical point for the Anderson transition. These results strongly suggest that the energy levels follow the Wigner-Dyson statistics in the extended phase, consistent with earlier analytical predictions for the Anderson model on an Erdös-Rényi random graph. Our results are obtained from the accurate numerical solution of an exact set of equations valid for infinitely large regular random graphs.