3 research outputs found
Temperature dependent fluctuations in the two-dimensional XY model
We present a detailed investigation of the probability density function (PDF)
of order parameter fluctuations in the finite two-dimensional XY (2dXY) model.
In the low temperature critical phase of this model, the PDF approaches a
universal non-Gaussian limit distribution in the limit T-->0. Our analysis
resolves the question of temperature dependence of the PDF in this regime, for
which conflicting results have been reported. We show analytically that a weak
temperature dependence results from the inclusion of multiple loop graphs in a
previously-derived graphical expansion. This is confirmed by numerical
simulations on two controlled approximations to the 2dXY model: the Harmonic
and ``Harmonic XY'' models. The Harmonic model has no
Kosterlitz-Thouless-Berezinskii (KTB) transition and the PDF becomes
progressively less skewed with increasing temperature until it closely
approximates a Gaussian function above T ~ 4\pi. Near to that temperature we
find some evidence of a phase transition, although our observations appear to
exclude a thermodynamic singularity.Comment: 15 pages, 5 figures and 1 tabl
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