Self-consistent equations and quantum diffusion for the Anderson model
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908634829553664 |
|---|---|
| author | Black, Adam Drogin, Reuben Hernández, Felipe |
| author_facet | Black, Adam Drogin, Reuben Hernández, Felipe |
| contents | We consider the Anderson tight-binding model on $\mathbb{Z}^d$, $d\geq 2$, with Gaussian noise and at low disorder $λ>0$. We derive a diffusive scaling limit for the entries of the resolvent $R(z)$ at imaginary part $\operatorname*{Im} z\simλ^{2+κ_d}$, $κ_d>0$, with high probability. As consequences, we establish quantum diffusion (in a time-averaged sense) for the Schrödinger propagator at the longest timescale known to date and improve the best available lower bounds on the localization length of eigenfunctions. Our results for $d=2$ are the first quantum diffusion results for the Anderson model on $\mathbb{Z}^2$. The proof avoids the use of diagrammatic expansions and instead proceeds by analyzing certain self-consistent equations for $R(z)$. This is facilitated by new estimates for $\|R(z)\|_{\ell^p\rightarrow \ell^q}$ that control the recollisions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06468 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Self-consistent equations and quantum diffusion for the Anderson model Black, Adam Drogin, Reuben Hernández, Felipe Mathematical Physics Analysis of PDEs Probability We consider the Anderson tight-binding model on $\mathbb{Z}^d$, $d\geq 2$, with Gaussian noise and at low disorder $λ>0$. We derive a diffusive scaling limit for the entries of the resolvent $R(z)$ at imaginary part $\operatorname*{Im} z\simλ^{2+κ_d}$, $κ_d>0$, with high probability. As consequences, we establish quantum diffusion (in a time-averaged sense) for the Schrödinger propagator at the longest timescale known to date and improve the best available lower bounds on the localization length of eigenfunctions. Our results for $d=2$ are the first quantum diffusion results for the Anderson model on $\mathbb{Z}^2$. The proof avoids the use of diagrammatic expansions and instead proceeds by analyzing certain self-consistent equations for $R(z)$. This is facilitated by new estimates for $\|R(z)\|_{\ell^p\rightarrow \ell^q}$ that control the recollisions. |
| title | Self-consistent equations and quantum diffusion for the Anderson model |
| topic | Mathematical Physics Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2506.06468 |