Functionals of the Brownian motion, localization and metric graphs

Alain Comtet 1, 2, Jean Desbois 1, Christophe Texier 1, 3

Journal of Physics A 38 (2005) R341-R383

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 Schr\\ödinger 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.

  • 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 2. Unite mixte de service de l'institut Henri Poincaré (UMSIHP),
    CNRS : UMS839 – Université Paris VI - Pierre et Marie Curie
  • 3. Laboratoire de Physique des Solides (LPS),
    CNRS : UMR8502 – Université Paris XI - Paris Sud