# Supersymmetric Polar Coordinates with applications to the Lloyd model

## TL;DR

This paper introduces supersymmetric polar coordinates to analyze spectral properties of random Schrödinger operators, providing a dual representation applicable to general distributions and applying it to the Lloyd model.

## Contribution

It develops a new supersymmetric polar coordinate method for dual representation of Green's functions, extending analysis to general distributions and specific models like the Lloyd model.

## Key findings

- Recovered the exact density of states for non-negative correlation in the Lloyd model.
- Showed the density of states is well approximated for small negative interactions.
- Results are uniform on the lattice  in volume.

## Abstract

Spectral properties of random Schr\"odinger operators are encoded in the average of products of Greens functions. For probability distributions with enough finite moments, the supersymmetric approach offers a useful dual representation. Here we use supersymmetric polar coordinates to derive a dual representation that holds for general distributions. We apply this result to study the density of states of the linearly correlated Lloyd model. In the case of non-negative correlation, we recover the well-known exact formula. In the case of linear small negative interaction localized around one point, we show that the density of states is well approximated by the exact formula. Our results hold on the lattice $\mathbb{Z}^d$ uniformly in the volume.