1,784 research outputs found
Poisson statistics of PageRank probabilities of Twitter and Wikipedia networks
We use the methods of quantum chaos and Random Matrix Theory for analysis of
statistical fluctuations of PageRank probabilities in directed networks. In
this approach the effective energy levels are given by a logarithm of PageRank
probability at a given node. After the standard energy level unfolding
procedure we establish that the nearest spacing distribution of PageRank
probabilities is described by the Poisson law typical for integrable quantum
systems. Our studies are done for the Twitter network and three networks of
Wikipedia editions in English, French and German. We argue that due to absence
of level repulsion the PageRank order of nearby nodes can be easily
interchanged. The obtained Poisson law implies that the nearby PageRank
probabilities fluctuate as random independent variables.Comment: 4 pages, 4 figures http://www.quantware.ups-tlse.fr
New Family of Solvable 1D Heisenberg Models
Starting from a Calogero--Sutherland model with hyperbolic interaction
confined by an external field with Morse potential we construct a Heisenberg
spin chain with exchange interaction on a lattice given
in terms of the zeroes of Laguerre polynomials. Varying the strength of the
Morse potential the Haldane--Shastry and harmonic spin chains are reproduced.
The spectrum of the models in this class is found to be that of a classical
one-dimensional Ising chain with nonuniform nearest neighbour coupling in a
nonuniform magnetic field which allows to study the thermodynamics in the limit
of infinite chains.Comment: 8 pp, LaTeX, ITP-UH-07/9
Breit-Wigner width for two interacting particles in one-dimensional random potential
For two interacting particles (TIP) in one-dimensional random potential the
dependence of the Breit-Wigner width , the local density of states and
the TIP localization length on system parameters is determined analytically.
The theoretical predictions for are confirmed by numerical
simulations.Comment: 10 pages Latex, 4 figures included. New version with extended
numerical results and discussions of earlier result
Dispersion relations and speeds of sound in special sectors for the integrable chain with alternating spins
Based on our previous analysis \cite{doerfel3} of the anisotropic integrable
chain consisting of spins and we compare the dispersion relations
for the sectors with infinite Fermi zones. Further we calculate the speeds of
sound for regions close to sector borders, where the Fermi radii either vanish
or diverge, and compare the results.Comment: 11 pages, LaTeX2e, uses iopart.cls,graphicx.sty and psfrag.sty, 2
figure
Level Statistics and Localization for Two Interacting Particles in a Random Potential
We consider two particles with a local interaction in a random potential
at a scale (the one particle localization length). A simplified
description is provided by a Gaussian matrix ensemble with a preferential
basis. We define the symmetry breaking parameter
associated to the statistical invariance under change of basis. We show that
the Wigner-Dyson rigidity of the energy levels is maintained up to an energy
. We find that when (the
inverse lifetime of the states of the preferential basis) is smaller than
(the level spacing), and when . This implies that the two-particle localization length first
increases as before eventually behaving as .Comment: 4 pages REVTEX, 4 Figures EPS, UUENCODE
--Oscillations for Correlated Electron Pairs in Disordered Mesoscopic Rings
The full spectrum of two interacting electrons in a disordered mesoscopic
one--dimensional ring threaded by a magnetic flux is calculated numerically.
For ring sizes far exceeding the one--particle localization length we
find several --periodic states whose eigenfunctions exhibit a pairing
effect. This represents the first direct observation of interaction--assisted
coherent pair propagation, the pair being delocalized on the scale of the whole
ring.Comment: 4 pages, uuencoded PostScript, containing 5 figures
Emergence of Quantum Ergodicity in Rough Billiards
By analytical mapping of the eigenvalue problem in rough billiards on to a
band random matrix model a new regime of Wigner ergodicity is found. There the
eigenstates are extended over the whole energy surface but have a strongly
peaked structure. The results of numerical simulations and implications for
level statistics are also discussed.Comment: revtex, 4 pages, 4 figure
Properties of the chiral spin liquid state in generalized spin ladders
We study zero temperature properties of a system of two coupled quantum spin
chains subject to fields explicitly breaking time reversal symmetry and parity.
Suitable choice of the strength of these fields gives a model soluble by Bethe
Ansatz methods which allows to determine the complete magnetic phase diagram of
the system and the asymptotics of correlation functions from the finite size
spectrum. The chiral properties of the system for both the integrable and the
nonintegrable case are studied using numerical techniques.Comment: 19 pages, 9eps figures, Late
Effective Model Formulation for Two Interacting Electrons in a Disordered Metal
We derive an analytical theory for two interacting electrons in a
--dimensional random potential. Our treatment is based on an effective
random matrix Hamiltonian. After mapping the problem on a nonlinear
model, we exploit similarities with the theory of disordered metals to identify
a scaling parameter, investigate the level correlation function, and study the
transport properties of the system. In agreement with recent numerical work we
find that pair propagation is subdiffusive and that the pair size grows
logarithmically with time.Comment: 4 pages, revtex, no figure
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