Localization and delocalization in the Anderson model
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three
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…
Full source abstract
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.
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Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder
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…
Full source 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.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
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