9 research outputs found
Absence of mobility edge for the Anderson random potential on tree graphs at weak disorder
Our recently established criterion for the formation of extended states on
tree graphs in the presence of disorder is shown to have the surprising
implication that for bounded random potentials, as in the Anderson model, there
is no transition to a spectral regime of Anderson localization, in the form
usually envisioned, unless the disorder is strong enough
Anderson localization on the Cayley tree : multifractal statistics of the transmission at criticality and off criticality
In contrast to finite dimensions where disordered systems display
multifractal statistics only at criticality, the tree geometry induces
multifractal statistics for disordered systems also off criticality. For the
Anderson tight-binding localization model defined on a tree of branching ratio
K=2 with generations, we consider the Miller-Derrida scattering geometry
[J. Stat. Phys. 75, 357 (1994)], where an incoming wire is attached to the root
of the tree, and where outcoming wires are attached to the leaves of
the tree. In terms of the transmission amplitudes , the total
Landauer transmission is , so that each channel
is characterized by the weight . We numerically measure the
typical multifractal singularity spectrum of these weights as a
function of the disorder strength and we obtain the following conclusions
for its left-termination point . In the delocalized phase ,
is strictly positive and is associated with a
moment index . At criticality, it vanishes and is
associated with the moment index . In the localized phase ,
is associated with some moment index . We discuss the
similarities with the exact results concerning the multifractal properties of
the Directed Polymer on the Cayley tree.Comment: v2=final version (16 pages
Resonances and Partial Delocalization on the Complete Graph
Random operators may acquire extended states formed from a multitude of
mutually resonating local quasi-modes. This mechanics is explored here in the
context of the random Schr\"odinger operator on the complete graph. The
operators exhibits local quasi modes mixed through a single channel. While most
of its spectrum consists of localized eigenfunctions, under appropriate
conditions it includes also bands of states which are delocalized in the
-though not in -sense, where the eigenvalues have the
statistics of \v{S}eba spectra. The analysis proceeds through some general
observations on the scaling limits of random functions in the Herglotz-Pick
class. The results are in agreement with a heuristic condition for the
emergence of resonant delocalization, which is stated in terms of the tunneling
amplitude among quasi-modes