Statistics of Green's functions on a disordered Cayley tree and the validity of forward scattering approximation

TL;DR

This paper investigates the accuracy of the forward scattering approximation for Green's functions in the Anderson model on a Cayley tree, deriving new approximations and analyzing their validity through supersymmetric methods.

Contribution

The paper establishes a relationship between Green's function moments and eigenvalues, introduces a new large-disorder approximation, and assesses the FSA's overestimation of resonance effects.

Findings

FSA overestimates the probability of large Green's functions.

The new approximation accurately predicts the eigenvalue in high disorder.

Error in FSA increases with distance between points.

Abstract

The accuracy of the forward scattering approximation for two-point Green's functions of the Anderson localization model on the Cayley tree is studied. A relationship between the moments of the Green's function and the largest eigenvalue of the linearized transfer-matrix equation is proved in the framework of the supersymmetric functional-integral method. The new large-disorder approximation for this eigenvalue is derived and its accuracy is established. Using this approximation the probability distribution of the two-point Green's function is found and compared with that in the forward scattering approximation (FSA). It is shown that FSA overestimates the role of resonances and thus the probability for the Green's function to be significantly larger than its typical value. The error of FSA increases with increasing the distance between points in a two-point Green's function.