Quantum Diffusion and the Block-Size Crossover in the Block Anderson Model
Abstract
We study the block Anderson Hamiltonian HL=λΨL+𝒱 on the discrete torus (ℤ/Lℤ)d, d≥2, where ΨL is the nearest-neighbor Laplacian and 𝒱 consists of independent Wd× Wd GOE blocks. In the weak-disorder limit λ→∞, we establish an entrywise local law, a spatial scaling limit for the squared resolvent entries, time-averaged dynamical delocalization, and lower bounds on the localization length of most bulk eigenfunctions. Particularly, we allow the block size to be fixed or grow above a polylogarithmic threshold. In sufficiently large polynomial volumes, the dynamical estimates hold for Cδ(λ+W/λ)≤ T≤ λ1+κdBA−δ, with a dimension-dependent exponent κdBA>0, and the spreading scale is √((λ3+λW)T). This identifies a transition at Wasympλ2 between λ3/2√T and √(λWT). The proof combines Gaussian moment estimates with deterministic stability of a matrix Dyson equation. A partial Bloch transform handles fixed blocks, while scalar reduction controls growing blocks. A Feshbach--Schur reduction of the block T-equation separates a single scalar diffusion mode from the stable internal cell modes. A renewal expansion and exact cell counting give a positive representation of the diffusion coefficient and control the fourth moment uniformly across the block-size transition. Notably, our argument does not require an enumeration of Feynman diagrams.
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BibTeX
@misc{fu2026quantum,
title = {{Quantum Diffusion and the Block-Size Crossover in the Block Anderson Model}},
author = {Haoxuan Fu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00598},
eprint = {2611.00598},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00598},
}
RIS
TY - UNPB
TI - Quantum Diffusion and the Block-Size Crossover in the Block Anderson Model
AU - Haoxuan Fu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00598
UR - https://arxiv.org/abs/2611.00598
AB - We study the block Anderson Hamiltonian H_L=λΨ_L+𝒱 on the discrete torus (ℤ/Lℤ)^d, d≥2, where Ψ_L is the nearest-neighbor Laplacian and 𝒱 consists of independent W^d× W^d GOE blocks. In the weak-disorder limit λ→∞, we establish an entrywise local law, a spatial scaling limit for the squared resolvent entries, time-averaged dynamical delocalization, and lower bounds on the localization length of most bulk eigenfunctions. Particularly, we allow the block size to be fixed or grow above a polylogarithmic threshold. In sufficiently large polynomial volumes, the dynamical estimates hold for C_δ(λ+W/λ)≤ T≤ λ^1+κ_d^BA−δ, with a dimension-dependent exponent κ_d^BA>0, and the spreading scale is √((λ^3+λW)T). This identifies a transition at Wasympλ^2 between λ^3/2√T and √(λWT). The proof combines Gaussian moment estimates with deterministic stability of a matrix Dyson equation. A partial Bloch transform handles fixed blocks, while scalar reduction controls growing blocks. A Feshbach--Schur reduction of the block T-equation separates a single scalar diffusion mode from the stable internal cell modes. A renewal expansion and exact cell counting give a positive representation of the diffusion coefficient and control the fourth moment uniformly across the block-size transition. Notably, our argument does not require an enumeration of Feynman diagrams.
ER -