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

Giuseppe Cannizzaro, Cyril Labbé, Willem van Zuijlen

TL;DR

This work analyzes the top of the spectrum for discrete Anderson Hamiltonians with correlated Gaussian potentials in growing volumes. It develops a mesoscopic splitting method that exploits decorrelation across well-separated boxes and a local deterministic model defined by a shape function and a fluctuation field, yielding a Poisson limit for the top eigenvalues and precise localisation around the corresponding noise maxima. A key finding is the subtle dependence of localisation centres on the near-origin covariance, captured by the parameter $\tau_L$; depending on its scaling, the relation between eigenfunction localisation and noise maxima can be direct or mediated by a decorated Poisson structure. The results unify into three regimes that mirror the classical i.i.d. Weibull/regularly varying pictures, and they provide a rigorous bridge between extreme value theory of Gaussian fields and spectral theory of random Schrödinger operators with correlated noise.

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.

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

TL;DR

This work analyzes the top of the spectrum for discrete Anderson Hamiltonians with correlated Gaussian potentials in growing volumes. It develops a mesoscopic splitting method that exploits decorrelation across well-separated boxes and a local deterministic model defined by a shape function and a fluctuation field, yielding a Poisson limit for the top eigenvalues and precise localisation around the corresponding noise maxima. A key finding is the subtle dependence of localisation centres on the near-origin covariance, captured by the parameter ; depending on its scaling, the relation between eigenfunction localisation and noise maxima can be direct or mediated by a decorated Poisson structure. The results unify into three regimes that mirror the classical i.i.d. Weibull/regularly varying pictures, and they provide a rigorous bridge between extreme value theory of Gaussian fields and spectral theory of random Schrödinger operators with correlated noise.

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 of the covariance function of the noise: in some situations, the largest eigenfunctions are not associated with the largest values of the noise.
Paper Structure (21 sections, 25 theorems, 262 equations)

This paper contains 21 sections, 25 theorems, 262 equations.

Table of Contents

  1. Introduction and main results
  2. The potential
  3. Main result
  4. Strategy of proof and structure of the article
  5. Notation and basic Gaussian estimates
  6. Local behaviour of correlated Gaussian fields
  7. Orthogonal decomposition and basic properties
  8. Deterministic shape around local maxima
  9. Tail distribution of the maximum
  10. Statistics of the maxima of correlated Gaussian fields
  11. Convergence of the maxima and fluctuations on $U_L$
  12. Proof of Theorem \ref{['Th:Potential']}
  13. Decorrelation estimates
  14. The mesoscopic eigenproblem
  15. Approximating the eigenproblem
  16. ...and 6 more sections

Key Result

Theorem 1.3

Under Assumption ass:dLaL, converges in law as $L\rightarrow \infty$ to a Poisson point process on $[-1,1]^d\times\mathbb{R}$ of intensity $\mathrm{d} x \otimes e^{-u} \mathrm{d} u$.

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.6: Eigenvalue order statistics
  • Theorem 1.7: Localisation
  • Theorem 1.8: Relationship with the maxima of $\xi_L$
  • Remark 1.9
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 45 more