Result 261, Mathematical physics

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.

Lean formalization Proof

The bigger picture

Why it matters

Randomness affects quantum motion differently in two dimensions than in higher dimensions. These manuscripts claim a sharp spectral contrast, showing how dimension changes the kinds of energy states available in a disordered lattice.

What changes?

An Anderson operator describes hopping between neighboring lattice sites with independent, uniformly distributed random site potentials. The manuscripts report, almost surely, pure-point spectrum throughout the spectrum in dimension two at each fixed positive disorder strength. In each fixed dimension at least three, each fixed sufficiently small positive disorder strength instead gives, almost surely, purely absolutely continuous spectrum on an open energy interval with nonzero spectral weight. The interval may depend on dimension but not disorder strength.

What does that help mathematicians do?

Pure-point spectrum means states can be expanded in normalizable energy eigenstates. Absolutely continuous spectrum means spectral weight has a density in energy. The reported results would therefore establish fundamentally different spectral structures: an eigenstate description across the full two-dimensional spectrum, but no normalizable eigenstates in the specified higher-dimensional interval. Nonzero weight ensures that interval is not spectrally empty. The higher-dimensional claim does not classify energies outside it.

Are there practical applications?

The immediate value is foundational for mathematical physics: the claims would give a rigorous spectral distinction between two-dimensional and higher-dimensional quantum lattice models with uniform random disorder. This bears on the mathematical study of localization and delocalization. The supplied statements concern spectral type, however, and do not themselves provide quantitative decay rates, transport rates, or predictions for a particular material.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three

September 23, 2026 93 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder

September 23, 2026 78 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/261.md.

Localization and delocalization in the Anderson model

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The linked formalization proves a supporting spectral statement for the nearest-neighbor Anderson operator on Z2\mathbb Z^2. For every disorder strength h>0h>0 with independent site potentials uniform on [−h,h][-h,h], it constructs the bounded self-adjoint operator almost surely and identifies its spectrum as the real interval [−4−h,4+h][-4-h,4+h].

This selected statement identifies the spectral set. It does not assert the pure-point spectral type claimed in the accompanying paper.

Comparator links

Result Comparator statement
Almost-sure spectrum of the planar Anderson operator PlanarAndersonSpectrum.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.