2025/08/18 by Shihe Liu, Yunfeng Shi, Liu, Shihe +3
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Probability (math.PR) #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2508.12714
openalex publication_date 2025/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.