2021/02/18 by Laure Dumaz, Cyril Labbé, Dumaz, Laure +1 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #Statistical Mechanics (cond-mat.stat-mech)
paper · doi:10.48550/arxiv.2102.09316
openalex publication_date 2021/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the behavior of the spectrum of the continuous Anderson Hamiltonian HL, with white noise potential, on a segment whose size L is sent to infinity. We zoom around energy levels E either of order 1 (Bulk regime) or of order 1≪ E ≪ L (Crossover regime). We show that the point process of (appropriately rescaled) eigenvalues and centers of mass converge to a Poisson point process. We also prove exponential localization of the eigenfunctions at an explicit rate. In addition, we show that the eigenfunctions converge to well-identified limits: in the Crossover regime, these limits are universal. Combined with the results of our companion paper arXiv:2102.05393, this identifies completely the transition between the localized and delocalized phases of the spectrum of HL. The two main technical challenges are the proof of a two-points or Minami estimate, as well as an estimate on the convergence to equilibrium of a hypoelliptic diffusion, the proof of which relies on Malliavin calculus and the theory of hypocoercivity.