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

Abstract

We calculate the level compressibility $\chi (W,L)$ 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 $\chi (W,L)$ approaches the limit ${\lim }_{L\to 0^{+}}\chi (W,L)=0$ 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.

7 pages, 3 figures

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