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

Giuseppe Cannizzaro; Cyril Labbé; Willem van Zuijlen

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arXiv · 2505.06051

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

Giuseppe Cannizzaro·Cyril Labbé·Willem van Zuijlen

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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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Giuseppe Cannizzaro, Cyril Labbé, Willem van Zuijlen. 2025-05-09. Top of the spectrum of discrete Anderson Hamiltonians with correlated Gaussian potentials. https://arxiv.org/abs/2505.06051

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