On localization for the alloy-type Anderson-Bernoulli model with long-range hopping
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| Format: | Preprint |
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2025
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| _version_ | 1866908493002309632 |
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| author | Liu, Shihe Shi, Yunfeng Zhang, Zhifei |
| author_facet | Liu, Shihe Shi, Yunfeng Zhang, Zhifei |
| contents | In this paper, we prove the Anderson localization near the spectral edge for some alloy-type Anderson-Bernoulli model on $\mathbb{Z}^d$ with exponential long-range hopping. This extends the work of Bourgain [Geometric Aspects of Functional Analysis, LNM 1850: 77--99, 2004], in which he pioneered a novel multi-scale analysis to treat Bernoulli random variables. Our proof is mainly based on Bourgain's method. However, to establish the initial scales Green's function estimates, we adapt the approach of Klopp [Comm. Math. Phys, Vol. 232, 125--155, 2002], which is based on the Floquet-Bloch theory and a certain quantitative uncertainty principle. Our proof also applies to an analogues model on $\mathbb{R}^d.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12714 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On localization for the alloy-type Anderson-Bernoulli model with long-range hopping Liu, Shihe Shi, Yunfeng Zhang, Zhifei Mathematical Physics Dynamical Systems Probability Spectral Theory In this paper, we prove the Anderson localization near the spectral edge for some alloy-type Anderson-Bernoulli model on $\mathbb{Z}^d$ with exponential long-range hopping. This extends the work of Bourgain [Geometric Aspects of Functional Analysis, LNM 1850: 77--99, 2004], in which he pioneered a novel multi-scale analysis to treat Bernoulli random variables. Our proof is mainly based on Bourgain's method. However, to establish the initial scales Green's function estimates, we adapt the approach of Klopp [Comm. Math. Phys, Vol. 232, 125--155, 2002], which is based on the Floquet-Bloch theory and a certain quantitative uncertainty principle. Our proof also applies to an analogues model on $\mathbb{R}^d.$ |
| title | On localization for the alloy-type Anderson-Bernoulli model with long-range hopping |
| topic | Mathematical Physics Dynamical Systems Probability Spectral Theory |
| url | https://arxiv.org/abs/2508.12714 |