21,356 research outputs found
Correspondence between many-particle excitations and the entanglement spectrum of disordered ballistic one-dimensional systems
Using exact diagonalization for non-interacting systems and density matrix
renormalization group for interacting systems we show that Li and Haldane's
conjecture on the correspondence between the low-lying many-particle excitation
spectrum and the entanglement spectrum holds for disordered ballistic
one-dimensional many-particle systems. In order to demonstrate the
correspondence we develop a computational efficient way to calculate the ES of
low-excitation of non-interacting systems. We observe and explain the presence
of an unexpected shell structure in the excitation structure. The low-lying
shell are robust and survive even for strong electron-electron interactions.Comment: 6 pages 4 figure
Extremal spacings between eigenphases of random unitary matrices and their tensor products
Extremal spacings between eigenvalues of random unitary matrices of size N
pertaining to circular ensembles are investigated. Explicit probability
distributions for the minimal spacing for various ensembles are derived for N =
4. We study ensembles of tensor product of k random unitary matrices of size n
which describe independent evolution of a composite quantum system consisting
of k subsystems. In the asymptotic case, as the total dimension N = n^k becomes
large, the nearest neighbor distribution P(s) becomes Poissonian, but
statistics of extreme spacings P(s_min) and P(s_max) reveal certain deviations
from the Poissonian behavior
Universal correlations and power-law tails in financial covariance matrices
Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical ensemble of asset-price covariances from a single instance of financial data sets. Local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets. They are in good agreement with the universal Tracy-Widom distribution and Wigner surmise, respectively.
This suggests a strong degree of robustness especially in the low-lying sector of the spectra, most relevant for portfolio selections.
Conversely, the global spectral density of a single covariance matrix as well as the average over all unfolded nearest-neighbour spacing distributions deviate from standard Gaussian random matrix predictions. The data are in fair agreement with a recently introduced generalised random matrix model, with correlations showing a power-law decay
Spectral statistics of random geometric graphs
We use random matrix theory to study the spectrum of random geometric graphs,
a fundamental model of spatial networks. Considering ensembles of random
geometric graphs we look at short range correlations in the level spacings of
the spectrum via the nearest neighbour and next nearest neighbour spacing
distribution and long range correlations via the spectral rigidity Delta_3
statistic. These correlations in the level spacings give information about
localisation of eigenvectors, level of community structure and the level of
randomness within the networks. We find a parameter dependent transition
between Poisson and Gaussian orthogonal ensemble statistics. That is the
spectral statistics of spatial random geometric graphs fits the universality of
random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert
and Watts-Strogatz random graph.Comment: 19 pages, 6 figures. Substantially updated from previous versio
Gaussian limits for generalized spacings
Nearest neighbor cells in , are used to define
coefficients of divergence (-divergences) between continuous multivariate
samples. For large sample sizes, such distances are shown to be asymptotically
normal with a variance depending on the underlying point density. In ,
this extends classical central limit theory for sum functions of spacings. The
general results yield central limit theorems for logarithmic -spacings,
information gain, log-likelihood ratios and the number of pairs of sample
points within a fixed distance of each other.Comment: Published in at http://dx.doi.org/10.1214/08-AAP537 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
An Efficient Coding Theory for a Dynamic Trajectory Predicts non-Uniform Allocation of Grid Cells to Modules in the Entorhinal Cortex
Grid cells in the entorhinal cortex encode the position of an animal in its
environment using spatially periodic tuning curves of varying periodicity.
Recent experiments established that these cells are functionally organized in
discrete modules with uniform grid spacing. Here we develop a theory for
efficient coding of position, which takes into account the temporal statistics
of the animal's motion. The theory predicts a sharp decrease of module
population sizes with grid spacing, in agreement with the trends seen in the
experimental data. We identify a simple scheme for readout of the grid cell
code by neural circuitry, that can match in accuracy the optimal Bayesian
decoder of the spikes. This readout scheme requires persistence over varying
timescales, ranging from ~1ms to ~1s, depending on the grid cell module. Our
results suggest that the brain employs an efficient representation of position
which takes advantage of the spatiotemporal statistics of the encoded variable,
in similarity to the principles that govern early sensory coding.Comment: 23 pages, 5 figures. Supplemental Information available from the
authors on request. A previous version of this work appeared in abstract form
(Program No. 727.02. 2015 Neuroscience Meeting Planner. Chicago, IL: Society
for Neuroscience, 2015. Online.
Energy levels and their correlations in quasicrystals
Quasicrystals can be considered, from the point of view of their electronic
properties, as being intermediate between metals and insulators. For example,
experiments show that quasicrystalline alloys such as AlCuFe or AlPdMn have
conductivities far smaller than those of the metals that these alloys are
composed from. Wave functions in a quasicrystal are typically intermediate in
character between the extended states of a crystal and the exponentially
localized states in the insulating phase, and this is also reflected in the
energy spectrum and the density of states. In the theoretical studies we
consider in this review, the quasicrystals are described by a pure hopping
tight binding model on simple tilings. We focus on spectral properties, which
we compare with those of other complex systems, in particular, the Anderson
model of a disordered metal.Comment: 15 pages including 19 figures. Review article, submitted to Phil. Ma
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