Diffusive mean-square displacement at fixed disorder
Let $(V_x)_{x \in \mathbb{Z}^3}$ be independent random variables, each uniform on $[-\sqrt{3}, \sqrt{3}]$, so of mean zero and variance one, and for $\lambda > 0$ let $H_\lambda$ be the Anderson model $$(H_\lambda \psi)(x) = \sum_{|y - x|_1 = 1} \psi(y) + \lambda V_x \psi(x)$$ on $\ell^2(\mathbb{Z}^3)$. Its mean-square displacement, averaged over the potential, is $$M_\lambda(t) = \mathbb{E}\Bigl[\sum_{x \in \mathbb{Z}^3} |x|_2^2 \, \bigl|\langle \delta_x, e^{-itH_\lambda} \delta_0 \rangle\bigr|^2\Bigr].$$ Quantum diffusion, $M_\lambda(t)$ growing linearly in $t$, is expected for small $\lambda$ in three dimensions. Erdős, Salmhofer and Yau proved, in every dimension $d \ge 3$, that in the scaling limit $\lambda \to 0$ with $t \sim \lambda^{-2-\kappa}$ and $x \sim \lambda^{-2-\kappa/2}$, for small $\kappa > 0$, the averaged Wigner distribution of the solution converges to a solution of a heat equation. For Gaussian potentials in dimensions $d \ge 2$, Black, Drogin and Hernández proved quantum diffusion in a time-averaged sense up to longer times, through a diffusive limit of the resolvent at imaginary part of order $\lambda^{2+\kappa_d}$.
Problem. Does there exist $\lambda_0 > 0$ such that, for every $0 < \lambda < \lambda_0$, there are constants $0 < c_\lambda \le C_\lambda < \infty$ and $t_\lambda < \infty$ with $$c_\lambda t \le M_\lambda(t) \le C_\lambda t \qquad (t \ge t_\lambda)?$$ Here $\lambda$ is held fixed as $t \to \infty$.
Source: Simon (2000), Problem 3, for ν ≥ 3 and disorder where there is absolutely continuous spectrum, with growth asymptotic to ct; with the average over the potential, Spencer (2012), §1.4. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- PR, SP
- Keywords
- AIM Open Applied Problems, Anderson model, quantum diffusion, random Schrödinger operators
References
@incollection{simon2000,
author = {Simon, Barry},
title = {Schr{\"o}dinger operators in the twenty-first century},
booktitle = {Mathematical Physics 2000},
editor = {Fokas, Athanassios and Grigoryan, Alexander and Kibble, Tom and Zegarlinski, Boguslaw},
publisher = {Imperial College Press},
address = {London},
year = {2000},
pages = {283--288},
doi = {10.1142/9781848160224_0014},
}
@article{spencer2012,
author = {Spencer, Thomas},
title = {Duality, statistical mechanics and random matrices},
journal = {Current Developments in Mathematics},
volume = {2012},
year = {2012},
pages = {229--260},
doi = {10.4310/CDM.2012.v2012.n1.a5},
}
@article{erdos2007,
author = {Erd{\H o}s, L{\'a}szl{\'o} and Salmhofer, Manfred and Yau, Horng-Tzer},
title = {Quantum diffusion for the {A}nderson model in the scaling limit},
journal = {Ann. Henri Poincar{\'e}},
volume = {8},
number = {4},
year = {2007},
pages = {621--685},
doi = {10.1007/s00023-006-0318-0},
eprint = {math-ph/0502025},
archivePrefix = {arXiv},
}
@misc{black2025,
author = {Black, Adam and Drogin, Reuben and Hern{\'a}ndez, Felipe},
title = {Self-consistent equations and quantum diffusion for the {A}nderson model},
year = {2025},
eprint = {2506.06468},
archivePrefix = {arXiv},
note = {Revised November 2025 (v2)},
}
@misc{aim-anderson-fixed-disorder-diffusion,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Diffusive mean-square displacement at fixed disorder}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/303-anderson-fixed-disorder-diffusion.md},
note = {Entry 303 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00326, "Diffusive mean-square displacement at fixed disorder", clopen.solutions.