103 research outputs found
Surprises in the phase diagram of the Anderson model on the Bethe lattice
The Anderson model on the Bethe lattice is historically among the first for
which an energy regime of extended states and a separate regime of localized
states could be established. In this paper, we review recently discovered
surprises in the phase diagram. Among them is that even at weak disorder, the
regime of diffusive transport extends well beyond energies of the unperturbed
model into the Lifshitz tails. As will be explained, the mechanism for the
appearance of extended states in this non-perturbative regime are
disorder-induced resonances. We also present remaining questions concerning the
structure of the eigenfunctions and the associated spectral statistics problem
on the Bethe lattice.Comment: Plenary lecture given at the International Congress on Mathematical
Physics, Aalborg, August 6-11, 201
On the ubiquity of the Cauchy distribution in spectral problems
We consider the distribution of the values at real points of random functions
which belong to the Herglotz-Pick (HP) class of analytic mappings of the upper
half plane into itself. It is shown that under mild stationarity assumptions
the individual values of HP functions with singular spectra have a Cauchy type
distribution. The statement applies to the diagonal matrix elements of random
operators, and holds regardless of the presence or not of level repulsion, i.e.
applies to both random matrix and Poisson-type spectra.Comment: Slightly revised version: presentation was made more explicit in
places, and additional references were provide
Kac-Ward formula and its extension to order-disorder correlators through a graph zeta function
A streamlined derivation of the Kac-Ward formula for the planar Ising model's
partition function is presented and applied in relating the kernel of the
Kac-Ward matrices' inverse with the correlation functions of the Ising model's
order-disorder correlation functions. A shortcut for both is facilitated by the
Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also
extended here to produce a family of non planar interactions on
for which the partition function and the order-disorder correlators are
solvable at special values of the coupling parameters/temperature.Comment: An extension of the Kac-Ward determinantal formula beyond planarity
was added (Section 5). To appear in Journal of Statistical Physic
Boosted Simon-Wolff Spectral Criterion and Resonant Delocalization
Discussed here are criteria for the existence of continuous components in the
spectra of operators with random potential. First, the essential condition for
the Simon-Wolff criterion is shown to be measurable at infinity. By
implication, for the iid case and more generally potentials with the K-property
the criterion is boosted by a zero-one law. The boosted criterion, combined
with tunneling estimates, is then applied for sufficiency conditions for the
presence of continuous spectrum for random Schr\"odinger operators. The general
proof strategy which this yields is modeled on the resonant delocalization
arguments by which continuous spectrum in the presence of disorder was
previously established for random operators on tree graphs. In another
application of the Simon-Wolff rank-one analysis we prove the almost sure
simplicity of the pure point spectrum for operators with random potentials of
conditionally continuous distribution.Comment: In version 2 the presentation was somewhat streamlined, and a related
new (/improved) result was added (Appendix B
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