# Anderson transition on the Bethe lattice: an approach with real energies

## TL;DR

This paper investigates the Anderson localization transition on the Bethe lattice using real energy propagators, introducing a new stability criterion, providing precise transition estimates, and connecting analytic approximations with asymptotic analysis.

## Contribution

It presents a novel stability-based criterion for localization transition, accurate transition point estimates, and links forward approximation with asymptotic results.

## Key findings

- New stability criterion for localization transition
- Precise numerical estimate of critical disorder
- Connection between forward approximation and asymptotic analysis

## Abstract

We study the Anderson model on the Bethe lattice by working directly with propagators at real energies $E$. We introduce a novel criterion for the localization-delocalization transition based on the stability of the population of the propagators, and show that it is consistent with the one obtained through the study of the imaginary part of the self-energy. We present an accurate numerical estimate of the transition point, as well as a concise proof of the asymptotic formula for the critical disorder on lattices of large connectivity, as given in [P.W. Anderson 1958]. We discuss how the forward approximation used in analytic treatments of localization problems fits into this scenario and how one can interpolate between it and the correct asymptotic analysis.