On localization for the alloy-type Anderson-Bernoulli model with long-range hopping

Key Points

Anderson localization occurs near the spectral edge in the alloy-type model, indicating complex behavior in random systems.
The key result derived focuses on green's function estimates crucial for initial scales in the long-range hopping model.
The approach adapts Bourgain’s multi-scale analysis along with Klopp’s methods based on floquet-bloch theory for robustness.
Implications of this work extend to other models in higher dimensions, pointing to broader applications in mathematical physics.

Abstract

In this paper, we prove the Anderson localization near the spectral edge for some alloy-type Anderson-Bernoulli model on Zᵈ 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 Rᵈ.

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Cite This Study

Liu et al. (Mon,) studied this question.

synapsesocial.com/papers/68d913a34ddcf71ba560bb0f https://doi.org/https://doi.org/10.48550/arxiv.2508.12714

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