1,525 research outputs found
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
Schur function averages for the real Ginibre ensemble
We derive an explicit simple formula for expectations of all Schur functions
in the real Ginibre ensemble. It is a positive integer for all entries of the
partition even and zero otherwise. The result can be used to determine the
average of any analytic series of elementary symmetric functions by Schur
function expansion
Probability distributions of Linear Statistics in Chaotic Cavities and associated phase transitions
We establish large deviation formulas for linear statistics on the
transmission eigenvalues of a chaotic cavity, in the framework of
Random Matrix Theory. Given any linear statistics of interest , the probability distribution of generically
satisfies the large deviation formula
, where
is a rate function that we compute explicitly in many cases
(conductance, shot noise, moments) and corresponds to different
symmetry classes. Using these large deviation expressions, it is possible to
recover easily known results and to produce new formulas, such as a closed form
expression for (where
) for arbitrary integer . The universal limit
is also computed exactly. The
distributions display a central Gaussian region flanked on both sides by
non-Gaussian tails. At the junction of the two regimes, weakly non-analytical
points appear, a direct consequence of phase transitions in an associated
Coulomb gas problem. Numerical checks are also provided, which are in full
agreement with our asymptotic results in both real and Laplace space even for
moderately small . Part of the results have been announced in [P. Vivo, S.N.
Majumdar and O. Bohigas, {\it Phys. Rev. Lett.} {\bf 101}, 216809 (2008)].Comment: 31 pages, 16 figures. To appear in Phys. Rev. B. Added section IVD
about comparison with other theories and numerical simulation
Long-range correlations in the wave functions of chaotic systems
We study correlations of the amplitudes of wave functions of a chaotic system
at large distances. For this purpose, a joint distribution function of the
amplitudes at two distant points in a sample is calculated analytically using
the supersymmetry technique. The result shows that, although in the limit of
the orthogonal and unitary symmetry classes the correlations vanish, they are
finite through the entire crossover regime and may be reduced only by
localization effects.Comment: 4 pages RevTex + 2 fig
Random matrix description of decaying quantum systems
This contribution describes a statistical model for decaying quantum systems
(e.g. photo-dissociation or -ionization). It takes the interference between
direct and indirect decay processes explicitely into account. The resulting
expressions for the partial decay amplitudes and the corresponding cross
sections may be considered a many-channel many-resonance generalization of
Fano's original work on resonance lineshapes [Phys. Rev 124, 1866 (1961)].
A statistical (random matrix) model is then introduced. It allows to describe
chaotic scattering systems with tunable couplings to the decay channels. We
focus on the autocorrelation function of the total (photo) cross section, and
we find that it depends on the same combination of parameters, as the
Fano-parameter distribution. These combinations are statistical variants of the
one-channel Fano parameter. It is thus possible to study Fano interference
(i.e. the interference between direct and indirect decay paths) on the basis of
the autocorrelation function, and thereby in the regime of overlapping
resonances. It allows us, to study the Fano interference in the limit of
strongly overlapping resonances, where we find a persisting effect on the level
of the weak localization correction.Comment: 16 pages, 2 figure
Hilbert--Schmidt volume of the set of mixed quantum states
We compute the volume of the convex N^2-1 dimensional set M_N of density
matrices of size N with respect to the Hilbert-Schmidt measure. The hyper--area
of the boundary of this set is also found and its ratio to the volume provides
an information about the complex structure of M_N. Similar investigations are
also performed for the smaller set of all real density matrices. As an
intermediate step we analyze volumes of the unitary and orthogonal groups and
of the flag manifolds.Comment: 13 revtex pages, ver 3: minor improvement
Universal lateral distribution of energy deposit in air showers and its application to shower reconstruction
The light intensity distribution in a shower image and its implications to
the primary energy reconstructed by the fluorescence technique are studied.
Based on detailed CORSIKA energy deposit simulations, a universal analytical
formula is derived for the lateral distribution of light in the shower image
and a correction factor is obtained to account for the fraction of shower light
falling into outlying pixels in the detector. The expected light profiles and
the corresponding correction of the primary shower energy are illustrated for
several typical event geometries. This correction of the shower energy can
exceed 10%, depending on shower geometry.Comment: 21 pages, 9 figure
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