413 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.
On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values
Given any fixed positive semi-definite diagonal matrix
we derive the explicit formula for the density of complex eigenvalues for
random matrices of the form } where the random unitary
matrices are distributed on the group according to the Haar
measure.Comment: 10 pages, 1 figur
Random Energy Model with complex replica number, complex temperatures and classification of the string's phases
The results by E. Gardner and B.Derrida have been enlarged for the complex
temperatures and complex numbers of replicas. The phase structure is found.
There is a connection with string models and their phase structure is analyzed
from the REM's point of view.Comment: 11 pages,revte
Induced vs Spontaneous Breakdown of S-matrix Unitarity: Probability of No Return in Quantum Chaotic and Disordered Systems
We investigate systematically sample-to sample fluctuations of the
probability of no return into a given entrance channel for wave
scattering from disordered systems. For zero-dimensional ("quantum chaotic")
and quasi one-dimensional systems with broken time-reversal invariance we
derive explicit formulas for the distribution of , and investigate
particular cases. Finally, relating to violation of S-matrix unitarity
induced by internal dissipation, we use the same quantity to identify the
Anderson delocalisation transition as the phenomenon of spontaneous breakdown
of S-matrix unitarity.Comment: This is the published version, with a few modifications added to the
last par
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
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
Pre-freezing of multifractal exponents in Random Energy Models with logarithmically correlated potential
Boltzmann-Gibbs measures generated by logarithmically correlated random
potentials are multifractal. We investigate the abrupt change ("pre-freezing")
of multifractality exponents extracted from the averaged moments of the measure
- the so-called inverse participation ratios. The pre-freezing can be
identified with termination of the disorder-averaged multifractality spectrum.
Naive replica limit employed to study a one-dimensional variant of the model is
shown to break down at the pre-freezing point. Further insights are possible
when employing zero-dimensional and infinite-dimensional versions of the
problem. In particular, the latter version allows one to identify the pattern
of the replica symmetry breaking responsible for the pre-freezing phenomenon.Comment: This is published version, 11 pages, 1 figur
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
Distribution of "level velocities" in quasi 1D disordered or chaotic systems with localization
The explicit analytical expression for the distribution function of
parametric derivatives of energy levels ("level velocities") with respect to a
random change of scattering potential is derived for the chaotic quantum
systems belonging to the quasi 1D universality class (quantum kicked rotator,
"domino" billiard, disordered wire, etc.).Comment: 11 pages, REVTEX 3.
Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of Noises generated by Gaussian Free Fields
We compute the distribution of the partition functions for a class of
one-dimensional Random Energy Models (REM) with logarithmically correlated
random potential, above and at the glass transition temperature. The random
potential sequences represent various versions of the 1/f noise generated by
sampling the two-dimensional Gaussian Free Field (2dGFF) along various planar
curves. Our method extends the recent analysis of Fyodorov Bouchaud from the
circular case to an interval and is based on an analytical continuation of the
Selberg integral. In particular, we unveil a {\it duality relation} satisfied
by the suitable generating function of free energy cumulants in the
high-temperature phase. It reinforces the freezing scenario hypothesis for that
generating function, from which we derive the distribution of extrema for the
2dGFF on the interval. We provide numerical checks of the circular and
the interval case and discuss universality and various extensions. Relevance to
the distribution of length of a segment in Liouville quantum gravity is noted.Comment: 25 pages, 12 figures Published version. Misprint corrected,
references and note adde
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