T-7

Goal: the goal of this set of problems is to derive an estimate for the transition point for the Anderson model on the Bethe lattice.
Techniques: cavity method, stability analysis.

A criterion for localization

Green functions and self-energies. Given a lattice $\Lambda$ with sites $a$ , we call $|a\rangle$ the wave function completely localised in site $a$ . The Anderson model has Hamiltonian $H=W\sum _{a}\epsilon _{a}|a\rangle \langle a|+\sum _{<a,b>}V_{ab}\left(|a\rangle \langle b|+|b\rangle \langle a|\right)=H_{0}+V$
where the local fields $\epsilon _{a}$ are random variables. We introduce the Green functions $G_{ab}(z)$ : these are functions of a complex variable belonging to the upper half of the complex plane, and are defined by [NOTA SU STILTJIES]
$G_{ab}(z)=\langle a|{\frac {1}{z-H}}|b\rangle$
The local self-energies $\sigma _{a}(z)$ are functions defined by the equality
$G_{aa}(z)=\langle a|{\frac {1}{z-H}}|a\rangle ={\frac {1}{z-\epsilon _{a}-\sigma _{a}(z)}}.$ It is clear that when the kinetic term $V$ in the Hamiltonian vanishes, the local self-energies vanish; these quantities encode how much the energy levels $\epsilon _{a}$ are shifted by the presence of the kinetic term $V$ . They are random functions, with a distribution that