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.
released 2026-09-23 | 1 theorem · 27 lemmas · 40 proofs · 41,173 words |
PLAY LEVEL 1 »(pdf)
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.
released 2026-09-23 | 3 theorems · 25 lemmas · 40 proofs · 34,800 words |
PLAY LEVEL 2 »(pdf)
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.