Math Conjectures

Anderson delocalization in dimension d≥3d\ge3

OPENLandmarkConjectureProposed 1958 · Canonical special case

Canonical statement

Let d≥3d\ge3 and
(Hλ,ωψ)(x)=−∑∣y−x∣=1ψ(y)+λωxψ(x)on ℓ2(Zd), (H_{\lambda,\omega}\psi)(x)= -\sum_{|y-x|=1}\psi(y)+\lambda\omega_x\psi(x) \quad\text{on }\ell^2(\mathbb Z^d),
where the ωx\omega_x are iid with a bounded compactly supported density that is positive near 00. For all sufficiently small λ>0\lambda>0, there is a nonempty open interval II in the interior of [−2d,2d][-2d,2d] on which Hλ,ωH_{\lambda,\omega} has almost surely nonempty purely absolutely continuous spectrum.
View source LaTeX
Let \(d\ge3\) and \[ (H_{\lambda,\omega}\psi)(x)= -\sum_{|y-x|=1}\psi(y)+\lambda\omega_x\psi(x) \quad\text{on }\ell^2(\mathbb Z^d), \] where the \(\omega_x\) are iid with a bounded compactly supported density that is positive near \(0\). For all sufficiently small \(\lambda>0\), there is a nonempty open interval \(I\) in the interior of \([-2d,2d]\) on which \(H_{\lambda,\omega}\) has almost surely nonempty purely absolutely continuous spectrum.
MATH-PHYS-003:[Anderson1958AbsenceDiffusion][AizenmanMolchanov1993Localization][Kirsch2008RandomSchrodinger][Stolz2011AndersonLocalization]
Mathematical physicsProbability theoryAnalysis and functional analysis

Notes

Anderson's 1958 analysis of diffusion in random lattices predicted that disorder can localize quantum particles [Anderson1958AbsenceDiffusion]; in three or more dimensions, a localization–delocalization transition is expected, with extended states surviving at weak disorder. In mathematical terms, for the discrete Schrödinger operator on ℓ2(Zd)\ell^2(\mathbb Z^d) with iid random potential of strength λ\lambda, the conjecture asserts that for d≥3d\ge3 and small λ\lambda there is an energy interval in the interior of the spectrum on which the spectrum is almost surely purely absolutely continuous — the spectral signature of extended states.

The localized side of the picture is on firm ground: localization is proved at large disorder and near spectral edges, notably by the fractional-moment method [AizenmanMolchanov1993Localization], and the theory is laid out in surveys [Kirsch2008RandomSchrodinger] [Stolz2011AndersonLocalization]. Delocalization, by contrast, is established only on trees and in mean-field or random-matrix analogues of the model.

For the standard iid lattice model on Zd\mathbb Z^d itself, no interval of absolutely continuous spectrum has been exhibited. The conjecture is open, and a resolution appears to require genuinely new tools for proving the existence of extended states in finite dimensions.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.