Anderson localization on Z² at arbitrary positive disorder
Follow the latest progress by email.
For the nearest-neighbor Anderson operator on ℓ²(Z²) with independent site potentials uniform on [−h,h], is the spectral type almost surely pure point with a complete eigenbasis for every fixed h>0?
Why it matters and what counts as progress
Why it matters: The all-energy, arbitrarily weak-disorder question extends localization results at strong disorder or spectral edges.
What counts as progress: A full pure-point theorem for every positive disorder strength, or a spectral counterexample for this law; stronger dynamical or decay estimates are separate conclusions.
How the problem developed
- 1958 · Theoretical prediction [1]
Philip W. Anderson Introduced the disorder-localization phenomenon for random lattices, motivating the study of spectral localization in lattice Schrödinger operators.
- 1979 · Theoretical prediction [2]
Elliott Abrahams, Philip W. Anderson, D. C. Licciardello and T. V. Ramakrishnan Scaling theory predicted absence of quantum diffusion for noninteracting disordered electrons in two dimensions.
- 1993 · Partial result [3]
Michael Aizenman and Stanislav Molchanov Proved localization at sufficiently large disorder or at extreme energies for broad classes of random lattice operators.
- 23 Sept 2026 · Reported claim [4]
OpenAI The preprint claims pure-point spectrum throughout the spectrum for the nearest-neighbour Z² Anderson model with iid uniform site potential at every fixed positive disorder strength.
AI contribution
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder
The OpenAI manuscript claims that the nearest-neighbor Anderson operator on ℓ²(Z²), with independent site potentials uniform on [−h,h], has almost surely pure-point spectral type and a complete eigenbasis for every fixed h>0. It also identifies the spectrum as [−4−h,4+h].
Sources
- Absence of diffusion in certain random lattices
- Scaling theory of localization: Absence of quantum diffusion in two dimensions
- Localization at large disorder and at extreme energies: An elementary derivation
- OpenAI two-dimensional Anderson pure-point manuscript
- Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder
- Lean scope for the Anderson spectral interval
- OpenAI mathematics release account
Website timeline
Publication dates and editorial changes.