18 research outputs found
Statistical distribution of quantum entanglement for a random bipartite state
We compute analytically the statistics of the Renyi and von Neumann entropies
(standard measures of entanglement), for a random pure state in a large
bipartite quantum system. The full probability distribution is computed by
first mapping the problem to a random matrix model and then using a Coulomb gas
method. We identify three different regimes in the entropy distribution, which
correspond to two phase transitions in the associated Coulomb gas. The two
critical points correspond to sudden changes in the shape of the Coulomb charge
density: the appearance of an integrable singularity at the origin for the
first critical point, and the detachement of the rightmost charge (largest
eigenvalue) from the sea of the other charges at the second critical point.
Analytical results are verified by Monte Carlo numerical simulations. A short
account of some of these results appeared recently in Phys. Rev. Lett. {\bf
104}, 110501 (2010).Comment: 7 figure
Random Convex Hulls and Extreme Value Statistics
In this paper we study the statistical properties of convex hulls of
random points in a plane chosen according to a given distribution. The points
may be chosen independently or they may be correlated. After a non-exhaustive
survey of the somewhat sporadic literature and diverse methods used in the
random convex hull problem, we present a unifying approach, based on the notion
of support function of a closed curve and the associated Cauchy's formulae,
that allows us to compute exactly the mean perimeter and the mean area enclosed
by the convex polygon both in case of independent as well as correlated points.
Our method demonstrates a beautiful link between the random convex hull problem
and the subject of extreme value statistics. As an example of correlated
points, we study here in detail the case when the points represent the vertices
of independent random walks. In the continuum time limit this reduces to
independent planar Brownian trajectories for which we compute exactly, for
all , the mean perimeter and the mean area of their global convex hull. Our
results have relevant applications in ecology in estimating the home range of a
herd of animals. Some of these results were announced recently in a short
communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special
issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting
Metal-insulator transition in two-dimensional disordered systems with power-law transfer terms
We investigate a disordered two-dimensional lattice model for noninteracting
electrons with long-range power-law transfer terms and apply the method of
level statistics for the calculation of the critical properties. The
eigenvalues used are obtained numerically by direct diagonalization. We find a
metal-insulator transition for a system with orthogonal symmetry. The exponent
governing the divergence of the correlation length at the transition is
extracted from a finite size scaling analysis and found to be . The critical eigenstates are also analyzed and the distribution of the
generalized multifractal dimensions is extrapolated.Comment: 4 pages with 4 figures, printed version: PRB, Rapid Communication
Statistical properties of power-law random banded unitary matrices in the delocalization-localization transition regime
Power-law random banded unitary matrices (PRBUM), whose matrix elements decay
in a power-law fashion, were recently proposed to model the critical statistics
of the Floquet eigenstates of periodically driven quantum systems. In this
work, we numerically study in detail the statistical properties of PRBUM
ensembles in the delocalization-localization transition regime. In particular,
implications of the delocalization-localization transition for the fractal
dimension of the eigenvectors, for the distribution function of the eigenvector
components, and for the nearest neighbor spacing statistics of the eigenphases
are examined. On the one hand, our results further indicate that a PRBUM
ensemble can serve as a unitary analog of the power-law random Hermitian matrix
model for Anderson transition. On the other hand, some statistical features
unseen before are found from PRBUM. For example, the dependence of the fractal
dimension of the eigenvectors of PRBUM upon one ensemble parameter displays
features that are quite different from that for the power-law random Hermitian
matrix model. Furthermore, in the time-reversal symmetric case the nearest
neighbor spacing distribution of PRBUM eigenphases is found to obey a
semi-Poisson distribution for a broad range, but display an anomalous level
repulsion in the absence of time-reversal symmetry.Comment: 10 pages + 13 fig
Localization of eigenvectors in random graphs
Using exact numerical diagonalization, we investigate localization in two classes of
random matrices corresponding to random graphs. The first class comprises the adjacency
matrices of Erdős-Rényi (ER) random graphs. The second one corresponds to random cubic
graphs, with Gaussian random variables on the diagonal. We establish the position of the
mobility edge, applying the finite-size analysis of the inverse participation ratio. The
fraction of localized states is rather small on the ER graphs and decreases when the
average degree increases. On the contrary, on cubic graphs the fraction of localized
states is large and tends to 1 when the strength of the disorder increases, implying that
for sufficiently strong disorder all states are localized. The distribution of the inverse
participation ratio in localized phase has finite width when the system size tends to
infinity and exhibits complicated multi-peak structure. We also confirm that the
statistics of level spacings is Poissonian in the localized regime, while for extended
states it corresponds to the Gaussian orthogonal ensemble