2 research outputs found
Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers
Some results on the ordered statistics of eigenvalues for one-dimensional
random Schr\"odinger Hamiltonians are reviewed. In the case of supersymmetric
quantum mechanics with disorder, the existence of low energy delocalized states
induces eigenvalue correlations and makes the ordered statistics problem
nontrivial. The resulting distributions are used to analyze the problem of
classical diffusion in a random force field (Sinai problem) in the presence of
weakly concentrated absorbers. It is shown that the slowly decaying averaged
return probability of the Sinai problem, \mean{P(x,t|x,0)}\sim \ln^{-2}t, is
converted into a power law decay, \mean{P(x,t|x,0)}\sim t^{-\sqrt{2\rho/g}},
where is the strength of the random force field and the density of
absorbers.Comment: 10 pages ; LaTeX ; 4 pdf figures ; Proceedings of the meeting
"Fundations and Applications of non-equilibrium statistical mechanics",
Nordita, Stockholm, october 2011 ; v2: appendix added ; v3: figure 2.left
adde
Functionals of the Brownian motion, localization and metric graphs
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\"odinger 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.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and
conclusion added. Several references adde