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Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three
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Abstract
For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤd with independent uniform site potentials almost surely has purely absolutely continuous spectrum with nonzero weight on an open energy interval. The interval may depend on d but is independent of the disorder strength. This settles the purely absolutely continuous energy-range question in Simon's Problem 1 for this model.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Vojkan Jakšić, Yoram Last, Extended States on the Bethe Lattice Revisited (arXiv:2610.11611, 8 Oct 2026). Gives a short proof of purely absolutely continuous spectrum on the Bethe lattice at weak disorder, and its Appendix B derives purity directly from the boundary-positivity assertion in equation (177) of this paper, leaving the finite-volume argument establishing that assertion unchanged.
- Patrick Lopatto (listed contributor), Mobility Edge for Sparse Complex Gaussian Matrices (hexagon:2610.00168v1, 12 Oct 2026). Claims a single mobility edge for the local limit of sparse complex Gaussian matrices (weighted Poisson–Galton–Watson trees) and cites both family papers in its introduction as recent lattice results. [AI-generated per source] [family-level]
Formalization
- Yao Xu (GitHub: cadamcat), anderson-ac-lean4 (GitHub repository, Lean 4 v4.34.1 / Mathlib, builds on the OpenAI
OAIlibrary at commit adc7f12). An independent Lean 4 development whose README says it proves a statement matching Theorem 1.1, phrased for uniform disorder on [a,b] with b−a<δ, withoutsorryor added axioms, and "developed with AI assistance"; these claims are as stated by the README. - OpenAI math release, Lean scope note for family 261 (mirror) (lean/docs/261.md). The family's linked formalization constructs the planar operator and identifies its almost-sure spectrum as $[-4-h,4+h]$, covering neither the planar pure-point claim nor this paper's higher-dimensional absolute-continuity claim. [family-level]
Related prior literature mentioned by others
- Vojkan Jakšić, Yoram Last, Spectral Structure of Anderson Type Hamiltonians (Inventiones Mathematicae 141, 561–577, 2000). Jakšić and Last's 2026 follow-up invokes Corollaries 1.1.1 and 1.1.3 of this work, also cited in this paper's bibliography, to shorten the passage from boundary positivity to spectral purity.
Possibly relevant literature (found by the agent, not checked)
- Simon Becker, Izak Oltman, Disorder on the hyperbolic square lattice I: Anderson delocalization and absolutely continuous spectrum (arXiv:2609.20798, 17 Sep 2026). Proves weak-disorder absolutely continuous spectrum, and the absence of singular spectrum on a set of positive measure, for the Anderson model on the product of Z^d with a hyperbolic square lattice, a non-tree setting relevant to the paper's comparison with earlier tree-only results.
Digests and expositions
- heyprof, Anderson localisation: 2D vs. 3D (3D) (interactive exposition, undated; accessed 2026-10-21). This family-specific visualization illustrates finite-lattice eigenstates and participation numbers, discusses finite-size limitations, and identifies the two manuscripts' spectral claims. [family-level]
News and blogs
- Azmat, Anderson Localization OpenAI Proof: How Extended States Survive Weak Disorder in 3D (Binary Verse AI, 2026-10-19). Outlines the claimed energy window, finite-volume certificate, adaptive potential revelations and spectral-averaging argument, distinguishing the spectral conclusion from a mobility-edge or diffusion theorem.
- Carlos Toledo, OpenAI's math release: what it claims beyond what it checks (audit report, findings dated 2026-10-10). The report lists family 261 among cases where it classifies the formalized statements at audited commit fd4aeeb2 as narrower than the announced results. [family-level]
Elsewhere
- Hexagon: Mobility Edge for Sparse Complex Gaussian Matrices (AI-generated; human understanding: some)
- VibeMathed: Problem record
- Emergent Mind: family 261
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026,
author = {{OpenAI}},
title = {{Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026/paper.pdf}{OAI:Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/261.md. The Lean statements have not been checked by people against the paper's claims.
Family 261: Localization and delocalization in the Anderson model
Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.
Other papers in this family: