Top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian potentials
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910963905593344 |
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| author | Cannizzaro, Giuseppe Labbé, Cyril van Zuijlen, Willem |
| author_facet | Cannizzaro, Giuseppe Labbé, Cyril van Zuijlen, Willem |
| contents | We investigate the top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian noise in the large volume limit. The class of Gaussian noises under consideration allows for long-range correlations. We show that the largest eigenvalues converge to a Poisson point process and we obtain a very precise description of the associated eigenfunctions near their localisation centres. We also relate these localisation centres with the locations of the maxima of the noise. Actually, our analysis reveals that this relationship depends in a subtle way on the behaviour near $0$ of the covariance function of the noise: in some situations, the largest eigenfunctions are not associated with the largest values of the noise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_06051 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian potentials Cannizzaro, Giuseppe Labbé, Cyril van Zuijlen, Willem Probability Primary 60H25, Secondary 82B44, 60G70 We investigate the top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian noise in the large volume limit. The class of Gaussian noises under consideration allows for long-range correlations. We show that the largest eigenvalues converge to a Poisson point process and we obtain a very precise description of the associated eigenfunctions near their localisation centres. We also relate these localisation centres with the locations of the maxima of the noise. Actually, our analysis reveals that this relationship depends in a subtle way on the behaviour near $0$ of the covariance function of the noise: in some situations, the largest eigenfunctions are not associated with the largest values of the noise. |
| title | Top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian potentials |
| topic | Probability Primary 60H25, Secondary 82B44, 60G70 |
| url | https://arxiv.org/abs/2505.06051 |