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Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder
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Abstract
For each fixed positive disorder strength, we prove that the nearest-neighbor Anderson operator on the square lattice with independent uniform site potentials almost surely has pure-point spectral type throughout its spectrum. This resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for the uniform single-site law.
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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
- 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
- OpenAI math release, Lean scope note for family 261 (mirror) (lean/docs/261.md, comparator
PlanarAndersonSpectrum.lean). The release's formalization constructs the operator almost surely and identifies its spectrum as $[-4-h,4+h]$ for every $h>0$; the note states that it does not assert the pure-point spectral type claimed in the paper. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Linjun Li, Shihe Liu, Lingfu Zhang, Localization near the edge for the lattice Anderson-Bernoulli model on general dimension (arXiv:2609.30209, v1 24 Sep 2026, v2 20 Oct 2026). Proves localization near the bottom of the spectrum for the Bernoulli Anderson model in every dimension d≥2, extending the edge-localization line (Ding–Smart, Hurtado) that the paper's introduction contrasts with its all-energy result.
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
- 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:Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026,
author = {{OpenAI}},
title = {{Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026/paper.pdf}{OAI:Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-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: