2005/04/30 by Alain Comtet, Jean Desbois, J. Desbois +1 · 2 citations
Mathematics · Physics and Astronomy · #Quantum chaos and dynamical systems #Random Matrices and Applications #Spectral Theory in Mathematical Physics #cond-mat.dis-nn #cond-mat.mes-hall #nlin.CD #quant-ph
paper · pdf · doi:10.1088/0305-4470/38/37/r01
published as J. Phys. A: Math. Gen. 38, R341-R383 (2005) · Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and conclusion added. Several references added
openalex publication_date 2005/08/31 · arxiv created 2005/10/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of the Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed : some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schrödinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of the planar Brownian motion.