Top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian potentials

2025/05/09 by Giuseppe Cannizzaro, Cyril Labbé, Cannizzaro, Giuseppe +3
Mathematics · Physics and Astronomy · #60G70 #FOS: Mathematics #Primary 60H25 #Probability (math.PR) #Quantum chaos and dynamical systems #Random Matrices and Applications #Secondary 82B44 #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.06051

openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

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