Strong-disorder expansion of the root-averaged density of states for the Anderson model on the Bethe lattice

TL;DR

This paper analyzes the root-averaged density of states for the Anderson model on the Bethe lattice under strong disorder, demonstrating analyticity and deriving a strong-disorder expansion of the density.

Contribution

It introduces a method combining random-walk expansion and complex analysis to establish analyticity and compute the strong-disorder expansion of the density of states.

Findings

The density of states is absolutely continuous and real analytic in the scaled energy window.

Odd coefficients in the expansion vanish, and higher coefficients relate to short closed walks on the tree.

Explicit first correction term computed for the uniform distribution.

Abstract

We study the root-averaged density of states for the Anderson model on the Bethe lattice in the strong-disorder regime. Here the density of states means the root-averaged spectral measure, not a finite-volume eigenvalue counting limit. We assume that the single-site distribution has compact support and has a locally analytic density on an interval $I^\sharp$ containing a given interval $I$. Combining the random-walk expansion on the tree with a complex-analytic argument for the single-site Stieltjes transforms, we prove that the scaled averaged diagonal resolvent has a holomorphic continuation to a complex neighborhood of $I$ for all sufficiently large $\lambda$. By the Stieltjes inversion formula, the root-averaged density of states measure is absolutely continuous on the scaled energy window $\lambda I$, and its density is real analytic and has a finite-order strong-disorder expansion there. In the scaled form $E=\lambda\xi$, the leading coefficient is the local density of the single-site distribution. All odd coefficients vanish, and the higher coefficients are finite sums determined by occupation profiles of short closed walks on the tree. For the uniform single-site distribution, we compute the first nonzero correction term explicitly.