Strong-disorder expansion of the root-averaged density of states for the Anderson model on the Bethe lattice
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♯ 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 λ. By the Stieltjes inversion formula, the root-averaged density of states measure is absolutely continuous on the scaled energy window λI, and its density is real analytic and has a finite-order strong-disorder expansion there. In the scaled form E=λξ, 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.
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BibTeX
@misc{kaminaga2026strongdisorder,
title = {{Strong-disorder expansion of the root-averaged density of states for the Anderson model on the Bethe lattice}},
author = {Masahiro Kaminaga},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.00478},
eprint = {2605.00478},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.00478},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strong-disorder expansion of the root-averaged density of states for the Anderson model on the Bethe lattice
AU - Masahiro Kaminaga
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2605.00478
UR - https://arxiv.org/abs/2605.00478
AB - 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^♯ 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 λ. By the Stieltjes inversion formula, the root-averaged density of states measure is absolutely continuous on the scaled energy window λI, and its density is real analytic and has a finite-order strong-disorder expansion there. In the scaled form E=λξ, 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.
ER -