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)

Manuscripts (2)

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

September 23, 2026 · source · cite as \cite{OAI:Absolutely-Continuous-Spectrum-for-Weak-Disorder-Anderson-Models-in-Dimensions-at-Least-Three-September-23-2026}

For each fixed dimension d ≥ 3 and each fixed sufficiently small positive disorder strength, we prove that the Anderson operator on ℤ^d 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.

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

September 23, 2026 · source · cite as \cite{OAI:Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026}

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.

Result 261 · Mathematical physics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.