Localization and delocalization in the Anderson model

Claims electrons stay trapped on a random square grid at any disorder, and can spread at weak disorder in 3D.

2 papers September 23, 2026 Explained by AI

The Anderson model describes an electron hopping on a grid where each site has a random energy. Physicists expect that in two dimensions any amount of randomness traps the electron, while in three dimensions weak randomness lets it spread on some energies. Two papers claim to prove both pictures for uniform random energies. Lean proves only that the allowed energies on the square grid fill the interval from −4 − h to 4 + h.

First step

1958

Philip W. Anderson, random site energies as a mechanism that suppresses quantum transport

Significance

Major

Planar localization at weak disorder and absolutely continuous spectrum in three dimensions are listed open problems from Simon's 2000 problem list.

Trust

Formal proof listed

Field

Mathematical physics

2 papers

The explanation

Picture an electron on a square grid, like a sheet of graph paper. At each step it can hop to one of its 4 neighbours. Each site x also has a random energy v_x, drawn uniformly between −h and h. The number h is the disorder strength. Without randomness, a wave spreading evenly across the grid has energy 2cos k₁ + 2cos k₂ for some wave numbers k₁ and k₂. Those values fill the interval from −4 to 4. Random energies widen this to −4 − h up to 4 + h. For h = 0.5 the allowed energies, called the spectrum, run from −4.5 to 4.5.

The question is whether the electron spreads or stays put. Anderson introduced random site energies as a way to suppress quantum transportSource 1. In maths language, staying put means pure-point spectrum: the grid has a full set of trapped wave shapes, called eigenvectors. Spreading means absolutely continuous spectrum, with no trapped states at those energies. A physical scaling theory predicts no true metal in two dimensionsSource 2. So in the plane, any disorder h > 0 should trap the electron.

Earlier proofs worked only for strong disorder or for energies near the edges. Fröhlich, Martinelli, Scoppola and Spencer showed pure-point spectrum in those regimesSource 3. Edge results remain restricted to a bottom interval of energiesSource 4. Weak disorder in the middle of the band is the hard case. There the randomness is a small push, and it must still beat the wave's tendency to spread.

The second paper claims the planar case. For each h > 0, almost surely the square-grid operator has pure-point spectrum and a complete basis of eigenvectorsSource 5. The paper says this resolves the pure-point part of the two-dimensional Anderson localization conjecture for uniform energiesSource 6. It does not claim exponential decay or dynamical localizationSource 7.

The first paper goes the other way, in dimension 3 and higher. There Simon's Problem 1 asks for an energy range with purely absolutely continuous spectrumSource 8. The paper claims such a range for small disorder: on a fixed open interval of energies, the spectrum is purely absolutely continuousSource 9. On a tree-shaped lattice, the Bethe lattice, Klein proved this decades earlierSource 10. In three dimensions the interval sits just above the bottom edge −6 of the free band. The paper does not locate the full mobility edge, the energy where spreading switches to trappingSource 11.

How sure is this? Lean, a computer proof checker, proves one supporting fact. On the square grid, the spectrum is almost surely the interval from −4 − h to 4 + h. The trapping and spreading claims are outside Lean. They rest on two long papers, and no specialist review of them has been published.

What it doesn't show. This does not prove exponential decay of eigenfunctions, dynamical localization, the location of the mobility edge, or quantum diffusion.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds. Lean proves only the spectral interval on the square grid. The trapping and spreading claims rest on two long papers with no published specialist review.

Words used

Anderson model: An electron hopping between neighbouring grid sites, where each site has its own random energy.
Disorder strength h: The width of the random energies: each site's energy is uniform between −h and h.
Spectrum: The set of energies the electron can have in this model.
Pure-point spectrum: The case where trapped wave shapes (eigenvectors) span everything; it means the electron stays localized.
Absolutely continuous spectrum: Energies with no trapped states, where waves spread out, like in a metal.
Almost surely: With probability one over the random choice of site energies.
Mobility edge: An energy where behaviour switches from spreading to trapped.

The original papers (2)

Lean notes

All quoted lines (11)

Anderson introduced random site energies as a mechanism for suppressing quantum transport, motivated by spin diffusion and impurity conduction.
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder Back to text
The conductance-scaling theory of Abrahams, Anderson, Licciardello and Ramakrishnan predicts the absence of true metallic behavior in two dimensions.
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder Back to text
Fr\"ohlich, Martinelli, Scoppola and Spencer subsequently established pure-point spectrum with exponentially decaying eigenfunctions for independent identically distributed diagonal potentials with a bounded density in sufficiently large-disorder and suitable tail-energy regimes.
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder Back to text
The latter localization theorems remain restricted to a bottom interval, even though their distributional assumptions are broader.
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder Back to text
For each fixed $h>0$ , almost surely $H_v$ has pure-point spectral type, admits a complete orthonormal eigenbasis, and has spectrum $\sigma(H_v)=[-4-h,4+h].$
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder Back to text
It resolves the pure-point assertion of the two-dimensional Anderson localization conjecture for a uniform single-site distribution, as formulated in.
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder Back to text
We do not assert one probability-one event for all disorder strengths, a bound uniform in energy, or a dynamical-localization theorem.
Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder Back to text
Simon's Problem 1 asks for a purely absolutely continuous energy range for the nearest-neighbor model with iid uniform potentials in dimensions at least three, at suitable disorder widths.
Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three Back to text
We prove a weak-disorder delocalization result for independent uniform site potentials in every fixed dimension $d\ge3$ : on a fixed open interval, the spectrum is purely absolutely continuous and has nonzero weight.
Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three Back to text
On the Bethe lattice, Klein proved purely absolutely continuous spectrum on compact intervals inside the free band at sufficiently weak disorder.
Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three Back to text
It does not locate the full mobility edge discussed after Problem 1 or resolve the diffusion question stated separately as Problem 3.
Absolutely Continuous Spectrum for Weak-Disorder Anderson Models in Dimensions at Least Three Back to text

Other sources

openai/math, Lean scope note for family 261 This selected statement identifies the spectral set. It does not assert the pure-point spectral type claimed in the accompanying paper.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

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