20,122 research outputs found
Universality in Complex Networks: Random Matrix Analysis
We apply random matrix theory to complex networks. We show that nearest
neighbor spacing distribution of the eigenvalues of the adjacency matrices of
various model networks, namely scale-free, small-world and random networks
follow universal Gaussian orthogonal ensemble statistics of random matrix
theory. Secondly we show an analogy between the onset of small-world behavior,
quantified by the structural properties of networks, and the transition from
Poisson to Gaussian orthogonal ensemble statistics, quantified by Brody
parameter characterizing a spectral property. We also present our analysis for
a protein-protein interaction network in budding yeast.Comment: 4+ pages, 4 figures, to appear in PRE, major change in the paper
including titl
Statistical properties of spectral fluctuations for a quantum system with infinitely many components
Extending the idea formulated in Makino {\it{et al}}[Phys.Rev.E
{\bf{67}},066205], that is based on the Berry--Robnik approach [M.V. Berry and
M. Robnik, J. Phys. A {\bf{17}}, 2413], we investigate the statistical
properties of a two-point spectral correlation for a classically integrable
quantum system. The eigenenergy sequence of this system is regarded as a
superposition of infinitely many independent components in the semiclassical
limit. We derive the level number variance (LNV) in the limit of infinitely
many components and discuss its deviations from Poisson statistics. The slope
of the limiting LNV is found to be larger than that of Poisson statistics when
the individual components have a certain accumulation. This property agrees
with the result from the semiclassical periodic-orbit theory that is applied to
a system with degenerate torus actions[D. Biswas, M.Azam,and S.V.Lawande, Phys.
Rev. A {\bf 43}, 5694].Comment: 6 figures, 10 page
Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models
We study the variance and the Laplace transform of the probability law of
linear eigenvalue statistics of unitary invariant Matrix Models of
n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test
function of statistics is smooth enough and using the asymptotic formulas by
Deift et al for orthogonal polynomials with varying weights, we show first that
if the support of the Density of States of the model consists of two or more
intervals, then in the global regime the variance of statistics is a
quasiperiodic function of n generically in the potential, determining the
model. We show next that the exponent of the Laplace transform of the
probability law is not in general 1/2variance, as it should be if the Central
Limit Theorem would be valid, and we find the asymptotic form of the Laplace
transform of the probability law in certain cases
A simple topological model with continuous phase transition
In the area of topological and geometric treatment of phase transitions and
symmetry breaking in Hamiltonian systems, in a recent paper some general
sufficient conditions for these phenomena in -symmetric systems
(i.e. invariant under reflection of coordinates) have been found out. In this
paper we present a simple topological model satisfying the above conditions
hoping to enlighten the mechanism which causes this phenomenon in more general
physical models. The symmetry breaking is testified by a continuous
magnetization with a nonanalytic point in correspondence of a critical
temperature which divides the broken symmetry phase from the unbroken one. A
particularity with respect to the common pictures of a phase transition is that
the nonanalyticity of the magnetization is not accompanied by a nonanalytic
behavior of the free energy.Comment: 17 pages, 7 figure
Level density and level-spacing distributions of random, self-adjoint, non-Hermitian matrices
We investigate the level-density and level-spacing distribution
of random matrices where is a (diagonal)
inner-product and is a random, real symmetric or complex Hermitian matrix
with independent entries drawn from a probability distribution with zero
mean and finite higher moments. Although not Hermitian, the matrix is
self-adjoint with respect to and thus has purely real eigenvalues. We find
that the level density is independent of the underlying
distribution , is solely characterized by , and therefore generalizes
Wigner's semicircle distribution . We find that the level-spacing
distributions are independent of , are dependent upon the
inner-product and whether is real or complex, and therefore generalize
the Wigner's surmise for level spacing. Our results suggest -dependent
generalizations of the well-known Gaussian Orthogonal Ensemble (GOE) and
Gaussian Unitary Ensemble (GUE) classes.Comment: 5 pages, 5 figures, revised tex
Magnetic susceptibility of ultra-small superconductor grains
For assemblies of superconductor nanograins, the magnetic response is
analyzed as a function of both temperature and magnetic field. In order to
describe the interaction energy of electron pairs for a huge number of
many-particle states, involved in calculations, we develop a simple
approximation, based on the Richardson solution for the reduced BCS Hamiltonian
and applicable over a wide range of the grain sizes and interaction strengths
at arbitrary distributions of single-electron energy levels in a grain. Our
study is focused upon ultra-small grains, where both the mean value of the
nearest-neighbor spacing of single-electron energy levels in a grain and
variations of this spacing from grain to grain significantly exceed the
superconducting gap in bulk samples of the same material. For these ultra-small
superconductor grains, the overall profiles of the magnetic susceptibility as a
function of magnetic field and temperature are demonstrated to be qualitatively
different from those for normal grains. We show that the analyzed signatures of
pairing correlations are sufficiently stable with respect to variations of the
average value of the grain size and its dispersion over an assembly of
nanograins. The presence of these signatures does not depend on a particular
choice of statistics, obeyed by single-electron energy levels in grains.Comment: 40 pages, 12 figures, submitted to Phys. Rev. B, E-mail addresses:
[email protected], [email protected], [email protected]
Distribution of interference in random quantum algorithms
We study the amount of interference in random quantum algorithms using a
recently derived quantitative measure of interference. To this end we introduce
two random circuit ensembles composed of random sequences of quantum gates from
a universal set, mimicking quantum algorithms in the quantum circuit
representation. We show numerically that these ensembles converge to the
well--known circular unitary ensemble (CUE) for general complex quantum
algorithms, and to the Haar orthogonal ensemble (HOE) for real quantum
algorithms. We provide exact analytical formulas for the average and typical
interference in the circular ensembles, and show that for sufficiently large
numbers of qubits a random quantum algorithm uses with probability close to one
an amount of interference approximately equal to the dimension of the Hilbert
space. As a by-product, we offer a new way of efficiently constructing random
operators from the Haar measures of CUE or HOE in a high dimensional Hilbert
space using universal sets of quantum gates.Comment: 14 pages revtex, 11 eps figure
Random matrices with external source and KP functions
In this paper we prove that the partition function in the random matrix model
with external source is a KP function.Comment: 12 pages, title change
Laminar Dispersion in Presence of Slip and Cbemical Reactions
Concentration profiles of a solute convectively diffusing and simultaneously undergoing homogeneous and heterogeneous chemical reactions under isothermal conditions in a non-Poisuelle laminar slip flow taking place in a closed parallel-plate channel with a permeable bed are presented. It is shown that presence of slip brings about an increase in concentration across the entire cross-section of the channel. Effect of varying reaction rate parameters on concentration profiles is also discussed
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