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PreprintarXiv licencearXiv

Localization near the edge for the lattice Anderson-Bernoulli model on general dimension

  • Linjun Li
  • Shihe Liu
  • Lingfu Zhang
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Abstract

The Anderson tight-binding model is a fundamental model of quantum transport and localization in disordered media. Solving a problem left open by Bourgain and Kenig, this paper proves Anderson localization near the bottom of the spectrum for the lattice Anderson model with Bernoulli potential, in any dimension d≥ 2. The proof uses the multiscale framework of Fröhlich-Spencer and Bourgain-Kenig, and the main new ingredient is a probabilistic discrete unique continuation principle (PDUC) for the discrete Schrödinger equation. This PDUC is established via a bootstrap argument and a key probabilistic lemma proved by adaptively revealing the random potential.

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Cite

BibTeXRIS
BibTeX
@misc{li2026localization,
  title = {{Localization near the edge for the lattice Anderson-Bernoulli model on general dimension}},
  author = {Linjun Li and Shihe Liu and Lingfu Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.30209},
  eprint = {2609.30209},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30209},
}
RIS
TY  - UNPB
TI  - Localization near the edge for the lattice Anderson-Bernoulli model on general dimension
AU  - Linjun Li
AU  - Shihe Liu
AU  - Lingfu Zhang
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.30209
UR  - https://arxiv.org/abs/2609.30209
AB  - The Anderson tight-binding model is a fundamental model of quantum transport and localization in disordered media. Solving a problem left open by Bourgain and Kenig, this paper proves Anderson localization near the bottom of the spectrum for the lattice Anderson model with Bernoulli potential, in any dimension d≥ 2. The proof uses the multiscale framework of Fröhlich-Spencer and Bourgain-Kenig, and the main new ingredient is a probabilistic discrete unique continuation principle (PDUC) for the discrete Schrödinger equation. This PDUC is established via a bootstrap argument and a key probabilistic lemma proved by adaptively revealing the random potential.
ER  -