# Fractional Moment Methods for Anderson Localization with SAW Representation

## TL;DR

This paper explores the fractional moment method (FMM) for Anderson localization, linking Green's functions to self-avoiding walks on graphs with bounded degree, offering new insights into localization phenomena.

## Contribution

It extends FMM to general graphs using self-avoiding walk representations, broadening the understanding of localization in disordered systems.

## Key findings

- FMM can be formulated using self-avoiding walks on graphs.
- Green's function properties relate to graph structure via self-avoiding walks.
- The method applies to graphs with bounded vertex degree.

## Abstract

The Green's function contains much information about physical systems. Mathematically, the fractional moment method (FMM) developed by Aizenman and Molchanov connects the Green's function and the transport of electrons in the Anderson model. Recently, it has been discovered that the Green's function on a graph can be represented using self-avoiding walks on a graph, which allows us to connect localization properties in the system and graph properties. We discuss FMM in terms of the self-avoiding walks on a general graph, the only general condition being that the graph has a uniform bound on the vertex degree.