On the Anderson transition on the Bethe lattice

Abstract

We study the Anderson model on the Bethe lattice by working directly with real energies \(E\). We show that the criterion for the stability of the populations leads to the same results as other, more traditional methods based on the imaginary part of the self energies for identifying the localized and delocalized phases. We present new numerical and analytical results, including an accurate finite-size scaling of the transition point, returning a critical length that diverges with exponent \(\nu=1/2\) in the delocalized region of the transition, as well as a concise proof of the asymptotic formula for the critical disorder, as given in [P.W. Anderson 1958]. We discuss how the forward approximation used in analytic treatments of MBL fits into this scenario and how one can interpolate between it and the correct asymptotic analysis.