Two-dimensional localization length scaling in the bulk

Determine whether, for the discrete Anderson Hamiltonian H = Δ + λV on Z2 with small coupling λ > 0, all bulk-energy eigenfunctions are localized with localization length of order exp(c λ^{-2}) for some constant c > 0, as predicted by scaling theory.

Background

For the Anderson model in d = 2, the scaling theory of localization predicts that bulk eigenfunctions remain localized but with very large localization lengths that scale exponentially in λ^{-2}. While rigorous results ensure localization at spectral edges in various dimensions, establishing the precise bulk behavior in two dimensions remains an open problem.

Confirming this prediction would clarify whether diffusion is ultimately suppressed in d = 2 over sufficiently long times and length scales, contrasting with the conjectured extended states in higher dimensions. The conjecture is noted as part of Simon’s problem list.