104 research outputs found
Statistical Properties of Random Banded Matrices with Strongly Fluctuating Diagonal Elements
The random banded matrices (RBM) whose diagonal elements fluctuate much
stronger than the off-diagonal ones were introduced recently by Shepelyansky as
a convenient model for coherent propagation of two interacting particles in a
random potential. We treat the problem analytically by using the mapping onto
the same supersymmetric nonlinear model that appeared earlier in
consideration of the standard RBM ensemble, but with renormalized parameters. A
Lorentzian form of the local density of states and a two-scale spatial
structure of the eigenfunctions revealed recently by Jacquod and Shepelyansky
are confirmed by direct calculation of the distribution of eigenfunction
components.Comment: 7 pages,RevTex, no figures Submitted to Phys.Rev.
Level curvature distribution in a model of two uncoupled chaotic subsystems
We study distributions of eigenvalue curvatures for a block diagonal random matrix perturbed by
a full random matrix. The most natural physical realization of this model is a quantum chaotic system
with some inherent symmetry, such that its energy levels form two independent subsequences,
subject to a generic perturbation which does not respect the symmetry. We describe analytically
a crossover in the form of a curvature distribution with a tunable parameter namely the ratio of
inter/intra subsystem coupling strengths. We find that the peak value of the curvature distribution
is much more sensitive to the changes in this parameter than the power law tail behaviour. This
observation may help to clarify some qualitative features of the curvature distributions observed
experimentally in acoustic resonances of quartz blocks
On the statistics of resonances and non-orthogonal eigenfunctions in a model for single-channel chaotic scattering
We describe analytical and numerical results on the statistical properties of
complex eigenvalues and the corresponding non-orthogonal eigenvectors for
non-Hermitian random matrices modeling one-channel quantum-chaotic scattering
in systems with broken time-reversal invariance.Comment: 4 pages, 2 figure
On absolute moments of characteristic polynomials of a certain class of complex random matrices
Integer moments of the spectral determinant of complex
random matrices are obtained in terms of the characteristic polynomial of
the Hermitian matrix for the class of matrices where is a
given matrix and is random unitary. This work is motivated by studies of
complex eigenvalues of random matrices and potential applications of the
obtained results in this context are discussed.Comment: 41 page, typos correcte
Localization and fluctuations of local spectral density on tree-like structures with large connectivity: Application to the quasiparticle line shape in quantum dots
We study fluctuations of the local density of states (LDOS) on a tree-like
lattice with large branching number . The average form of the local spectral
function (at given value of the random potential in the observation point)
shows a crossover from the Lorentzian to semicircular form at ,
where , is the typical value of the hopping matrix
element, and is the width of the distribution of random site energies. For
the LDOS fluctuations (with respect to this average form) are
weak. In the opposite case, , the fluctuations get strong and the
average LDOS ceases to be representative, which is related to the existence of
the Anderson transition at . On the localized side
of the transition the spectrum is discrete, and LDOS is given by a set of
-like peaks. The effective number of components in this regime is given
by , with being the inverse participation ratio. It is shown that
has in the transition point a limiting value close to unity, , so that the system undergoes a transition directly from the deeply
localized to extended phase. On the side of delocalized states, the peaks in
LDOS get broadened, with a width being exponentially small near the
transition point. We discuss application of our results to the problem of the
quasiparticle line shape in a finite Fermi system, as suggested recently by
Altshuler, Gefen, Kamenev, and Levitov.Comment: 12 pages, 1 figure. Misprints in eqs.(21) and (28) corrected, section
VII added. Accepted for publication in Phys. Rev.
Exact asymptotics of the freezing transition of a logarithmically correlated random energy model
We consider a logarithmically correlated random energy model, namely a model
for directed polymers on a Cayley tree, which was introduced by Derrida and
Spohn. We prove asymptotic properties of a generating function of the partition
function of the model by studying a discrete time analogy of the KPP-equation -
thus translating Bramson's work on the KPP-equation into a discrete time case.
We also discuss connections to extreme value statistics of a branching random
walk and a rescaled multiplicative cascade measure beyond the critical point
Delay times and reflection in chaotic cavities with absorption
Absorption yields an additional exponential decay in open quantum systems
which can be described by shifting the (scattering) energy E along the
imaginary axis, E+i\hbar/2\tau_{a}. Using the random matrix approach, we
calculate analytically the distribution of proper delay times (eigenvalues of
the time-delay matrix) in chaotic systems with broken time-reversal symmetry
that is valid for an arbitrary number of generally nonequivalent channels and
an arbitrary absorption rate 1/\tau_{a}. The relation between the average delay
time and the ``norm-leakage'' decay function is found. Fluctuations above the
average at large values of delay times are strongly suppressed by absorption.
The relation of the time-delay matrix to the reflection matrix S^{\dagger}S is
established at arbitrary absorption that gives us the distribution of
reflection eigenvalues. The particular case of single-channel scattering is
explicitly considered in detail.Comment: 5 pages, 3 figures; final version to appear in PRE (relation to
reflection extended, new material with Fig.3 added, experiment
cond-mat/0305090 discussed
Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal noise
To understand the sample-to-sample fluctuations in disorder-generated
multifractal patterns we investigate analytically as well as numerically the
statistics of high values of the simplest model - the ideal periodic
Gaussian noise. By employing the thermodynamic formalism we predict the
characteristic scale and the precise scaling form of the distribution of number
of points above a given level. We demonstrate that the powerlaw forward tail of
the probability density, with exponent controlled by the level, results in an
important difference between the mean and the typical values of the counting
function. This can be further used to determine the typical threshold of
extreme values in the pattern which turns out to be given by
with . Such observation provides a
rather compelling explanation of the mechanism behind universality of .
Revealed mechanisms are conjectured to retain their qualitative validity for a
broad class of disorder-generated multifractal fields. In particular, we
predict that the typical value of the maximum of intensity is to be
given by , where is the
corresponding singularity spectrum vanishing at . For the
noise we also derive exact as well as well-controlled approximate
formulas for the mean and the variance of the counting function without
recourse to the thermodynamic formalism.Comment: 28 pages; 7 figures, published version with a few misprints
corrected, editing done and references adde
Imaginary Potential as a Counter of Delay Time for Wave Reflection from a 1D Random Potential
We show that the delay time distribution for wave reflection from a
one-dimensional random potential is related directly to that of the reflection
coefficient, derived with an arbitrarily small but uniform imaginary part added
to the random potential. Physically, the reflection coefficient, being
exponential in the time dwelt in the presence of the imaginary part, provides a
natural counter for it. The delay time distribution then follows
straightforwardly from our earlier results for the reflection coefficient, and
coincides with the distribution obtained recently by Texier and Comtet
[C.Texier and A. Comtet, Phys.Rev.Lett. {\bf 82}, 4220 (1999)],with all moments
infinite. Delay time distribution for a random amplifying medium is then
derived . In this case, however, all moments work out to be finite.Comment: 4 pages, RevTeX, replaced with added proof, figure and references. To
appear in Phys. Rev. B Jan01 200
On random symmetric matrices with a constraint: the spectral density of random impedance networks
We derive the mean eigenvalue density for symmetric Gaussian random N x N
matrices in the limit of large N, with a constraint implying that the row sum
of matrix elements should vanish. The result is shown to be equivalent to a
result found recently for the average density of resonances in random impedance
networks [Y.V. Fyodorov, J. Phys. A: Math. Gen. 32, 7429 (1999)]. In the case
of banded matrices, the analytical results are compared with those extracted
from the numerical solution of Kirchhoff equations for quasi one-dimensional
random impedance networks.Comment: 4 pages, 5 figure
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