1,767 research outputs found
Fluctuations and Ergodicity of the Form Factor of Quantum Propagators and Random Unitary Matrices
We consider the spectral form factor of random unitary matrices as well as of
Floquet matrices of kicked tops. For a typical matrix the time dependence of
the form factor looks erratic; only after a local time average over a suitably
large time window does a systematic time dependence become manifest. For
matrices drawn from the circular unitary ensemble we prove ergodicity: In the
limits of large matrix dimension and large time window the local time average
has vanishingly small ensemble fluctuations and may be identified with the
ensemble average. By numerically diagonalizing Floquet matrices of kicked tops
with a globally chaotic classical limit we find the same ergodicity. As a
byproduct we find that the traces of random matrices from the circular
ensembles behave very much like independent Gaussian random numbers. Again,
Floquet matrices of chaotic tops share that universal behavior. It becomes
clear that the form factor of chaotic dynamical systems can be fully faithful
to random-matrix theory, not only in its locally time-averaged systematic time
dependence but also in its fluctuations.Comment: 12 pages, RevTEX, 4 figures in eps forma
Nonlinear statistics of quantum transport in chaotic cavities
Copyright © 2008 The American Physical Society.In the framework of the random matrix approach, we apply the theory of Selberg’s integral to problems of quantum transport in chaotic cavities. All the moments of transmission eigenvalues are calculated analytically up to the fourth order. As a result, we derive exact explicit expressions for the skewness and kurtosis of the conductance and transmitted charge as well as for the variance of the shot-noise power in chaotic cavities. The obtained results are generally valid at arbitrary numbers of propagating channels in the two attached leads. In the particular limit of large (and equal) channel numbers, the shot-noise variance attends the universal value 1∕64β that determines a universal Gaussian statistics of shot-noise fluctuations in this case.DFG and BRIEF
Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry
The statistical properties of quantum transport through a chaotic cavity are
encoded in the traces \T={\rm Tr}(tt^\dag)^n, where is the transmission
matrix. Within the Random Matrix Theory approach, these traces are random
variables whose probability distribution depends on the symmetries of the
system. For the case of broken time-reversal symmetry, we present explicit
closed expressions for the average value and for the variance of \T for all
. In particular, this provides the charge cumulants \Q of all orders. We
also compute the moments of the conductance . All the
results obtained are exact, {\it i.e.} they are valid for arbitrary numbers of
open channels.Comment: 5 pages, 4 figures. v2-minor change
How often is a random quantum state k-entangled?
The set of trace preserving, positive maps acting on density matrices of size
d forms a convex body. We investigate its nested subsets consisting of
k-positive maps, where k=2,...,d. Working with the measure induced by the
Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes
of these sets. Our results strongly suggest that the inner set of
(k+1)-positive maps forms a small fraction of the outer set of k-positive maps.
These results are related to analogous bounds for the relative volume of the
sets of k-entangled states describing a bipartite d X d system.Comment: 19 pages in latex, 1 figure include
Thermodynamic Properties and Phase Transitions in a Mean-Field Ising Spin Glass on Lattice Gas: the Random Blume-Emery-Griffiths-Capel Model
The study of the mean-field static solution of the Random
Blume-Emery-Griffiths-Capel model, an Ising-spin lattice gas with quenched
random magnetic interaction, is performed. The model exhibits a paramagnetic
phase, described by a stable Replica Symmetric solution. When the temperature
is decreased or the density increases, the system undergoes a phase transition
to a Full Replica Symmetry Breaking spin-glass phase. The nature of the
transition can be either of the second order (like in the
Sherrington-Kirkpatrick model) or, at temperature below a given critical value,
of the first order in the Ehrenfest sense, with a discontinuous jump of the
order parameter and accompanied by a latent heat. In this last case coexistence
of phases takes place. The thermodynamics is worked out in the Full Replica
Symmetry Breaking scheme, and the relative Parisi equations are solved using a
pseudo-spectral method down to zero temperature.Comment: 24 pages, 12 figure
Skew-orthogonal Laguerre polynomials for chiral real asymmetric random matrices
We apply the method of skew-orthogonal polynomials (SOP) in the complex plane to asymmetric random matrices with real elements, belonging to two different classes. Explicit integral representations valid for arbitrary weight functions are derived for the SOP and for their Cauchy transforms, given as expectation values of traces and determinants or their inverses, respectively. Our proof uses the fact that the joint probability distribution function for all combinations of real eigenvalues and complex conjugate eigenvalue pairs can be written as a product. Examples for the SOP are given in terms of Laguerre polynomials for the chiral ensemble (also called the non-Hermitian real Wishart-Laguerre ensemble), both without and with the insertion of characteristic polynomials. Such characteristic polynomials play the role of mass terms in applications to complex Dirac spectra in field theory. In addition, for the elliptic real Ginibre ensemble we recover the SOP of Forrester and Nagao in terms of Hermite polynomials
Statistical properties of random density matrices
Statistical properties of ensembles of random density matrices are
investigated. We compute traces and von Neumann entropies averaged over
ensembles of random density matrices distributed according to the Bures
measure. The eigenvalues of the random density matrices are analyzed: we derive
the eigenvalue distribution for the Bures ensemble which is shown to be broader
then the quarter--circle distribution characteristic of the Hilbert--Schmidt
ensemble. For measures induced by partial tracing over the environment we
compute exactly the two-point eigenvalue correlation function.Comment: 8 revtex pages with one eps file included, ver. 2 - minor misprints
correcte
The 3-SAT problem with large number of clauses in -replica symmetry breaking scheme
In this paper we analyze the structure of the UNSAT-phase of the
overconstrained 3-SAT model by studying the low temperature phase of the
associated disordered spin model. We derive the Replica Symmetry
Broken equations for a general class of disordered spin models which includes
the Sherrington - Kirkpatrick model, the Ising -spin model as well as the
overconstrained 3-SAT model as particular cases. We have numerically solved the
Replica Symmetry Broken equations using a pseudo-spectral code down to
and including zero temperature. We find that the UNSAT-phase of the
overconstrained 3-SAT model is of the -RSB kind: in order to get a
stable solution the replica symmetry has to be broken in a continuous way,
similarly to the SK model in external magnetic field.Comment: 19 pages, 7 figures; some section improved; iopart styl
Inference of kinetic Ising model on sparse graphs
Based on dynamical cavity method, we propose an approach to the inference of
kinetic Ising model, which asks to reconstruct couplings and external fields
from given time-dependent output of original system. Our approach gives an
exact result on tree graphs and a good approximation on sparse graphs, it can
be seen as an extension of Belief Propagation inference of static Ising model
to kinetic Ising model. While existing mean field methods to the kinetic Ising
inference e.g., na\" ive mean-field, TAP equation and simply mean-field, use
approximations which calculate magnetizations and correlations at time from
statistics of data at time , dynamical cavity method can use statistics of
data at times earlier than to capture more correlations at different time
steps. Extensive numerical experiments show that our inference method is
superior to existing mean-field approaches on diluted networks.Comment: 9 pages, 3 figures, comments are welcom
- …