30 research outputs found
Non-universality of compact support probability distributions in random matrix theory
The two-point resolvent is calculated in the large-n limit for the generalized fixed and bounded trace ensembles. It is shown to disagree with that of the canonical Gaussian ensemble by a nonuniversal part that is given explicitly for all monomial potentials V(M)=M2p. Moreover, we prove that for the generalized fixed and bounded trace ensemble all k-point resolvents agree in the large-n limit, despite their nonuniversality
Real symmetric random matrices and paths counting
Exact evaluation of is here performed for real symmetric
matrices of arbitrary order , up to some integer , where the matrix
entries are independent identically distributed random variables, with an
arbitrary probability distribution.
These expectations are polynomials in the moments of the matrix entries ;
they provide useful information on the spectral density of the ensemble in the
large limit. They also are a straightforward tool to examine a variety of
rescalings of the entries in the large limit.Comment: 23 pages, 10 figures, revised pape
On high moments of strongly diluted large Wigner random matrices
We consider a dilute version of the Wigner ensemble of nxn random matrices
and study the asymptotic behavior of their moments in the limit of
infinite , and , where is the dilution parameter. We show
that in the asymptotic regime of the strong dilution, the moments with
depend on the second and the fourth moments of the random entries
and do not depend on other even moments of . This fact can be
regarded as an evidence of a new type of the universal behavior of the local
eigenvalue distribution of strongly dilute random matrices at the border of the
limiting spectrum. As a by-product of the proof, we describe a new kind of
Catalan-type numbers related with the tree-type walks.Comment: 43 pages (version four: misprints corrected, discussion added, other
minor modifications
The integrated density of states of the random graph Laplacian
We analyse the density of states of the random graph Laplacian in the
percolating regime. A symmetry argument and knowledge of the density of states
in the nonpercolating regime allows us to isolate the density of states of the
percolating cluster (DSPC) alone, thereby eliminating trivially localised
states due to finite subgraphs. We derive a nonlinear integral equation for the
integrated DSPC and solve it with a population dynamics algorithm. We discuss
the possible existence of a mobility edge and give strong evidence for the
existence of discrete eigenvalues in the whole range of the spectrum.Comment: 4 pages, 1 figure. Supplementary material available at
http://www.theorie.physik.uni-goettingen.de/~aspel/data/spectrum_supplement.pd
Probability density of determinants of random matrices
In this brief paper the probability density of a random real, complex and
quaternion determinant is rederived using singular values. The behaviour of
suitably rescaled random determinants is studied in the limit of infinite order
of the matrices
Random matrix theory for CPA: Generalization of Wegner's --orbital model
We introduce a generalization of Wegner's -orbital model for the
description of randomly disordered systems by replacing his ensemble of
Gaussian random matrices by an ensemble of randomly rotated matrices. We
calculate the one- and two-particle Green's functions and the conductivity
exactly in the limit . Our solution solves the CPA-equation of the
-Anderson model for arbitrarily distributed disorder. We show how the
Lloyd model is included in our model.Comment: 3 pages, Rev-Te
Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices
We study the fluctuations of the matrix entries of regular functions of
Wigner random matrices in the limit when the matrix size goes to infinity. In
the case of the Gaussian ensembles (GOE and GUE) this problem was considered by
A.Lytova and L.Pastur in J. Stat. Phys., v.134, 147-159 (2009). Our results are
valid provided the off-diagonal matrix entries have finite fourth moment, the
diagonal matrix entries have finite second moment, and the test functions have
four continuous derivatives in a neighborhood of the support of the Wigner
semicircle law.Comment: minor corrections; the manuscript will appear in the Journal of
Statistical Physic
Spectral form factor in a random matrix theory
In the theory of disordered systems the spectral form factor , the
Fourier transform of the two-level correlation function with respect to the
difference of energies, is linear for and constant for
. Near zero and near its exhibits oscillations which have
been discussed in several recent papers. In the problems of mesoscopic
fluctuations and quantum chaos a comparison is often made with random matrix
theory. It turns out that, even in the simplest Gaussian unitary ensemble,
these oscilllations have not yet been studied there. For random matrices, the
two-level correlation function exhibits several
well-known universal properties in the large N limit. Its Fourier transform is
linear as a consequence of the short distance universality of
. However the cross-over near zero and
requires to study these correlations for finite N. For this purpose we use an
exact contour-integral representation of the two-level correlation function
which allows us to characterize these cross-over oscillatory properties. The
method is also extended to the time-dependent case.Comment: 36P, (+5 figures not included
Spectrum and diffusion for a class of tight-binding models on hypercubes
We propose a class of exactly solvable anisotropic tight-binding models on an
infinite-dimensional hypercube. The energy spectrum is analytically computed
and is shown to be fractal and/or absolutely continuous according to the value
hopping parameters. In both cases, the spectral and diffusion exponents are
derived. The main result is that, even if the spectrum is absolutely
continuous, the diffusion exponent for the wave packet may be anything between
0 and 1 depending upon the class of models.Comment: 5 pages Late
A generalized spherical version of the Blume-Emery-Griffits model with ferromagnetic and antiferromagnetic interactions
We have investigated analitycally the phase diagram of a generalized
spherical version of the Blume-Emery-Griffiths model that includes
ferromagnetic or antiferromagnetic spin interactions as well as quadrupole
interactions in zero and nonzero magnetic field. We show that in three
dimensions and zero magnetic field a regular paramagnetic-ferromagnetic (PM-FM)
or a paramagnetic-antiferromagnetic (PM-AFM) phase transition occurs whenever
the magnetic spin interactions dominate over the quadrupole interactions.
However, when spin and quadrupole interactions are important, there appears a
reentrant FM-PM or AFM-PM phase transition at low temperatures, in addition to
the regular PM-FM or PM-AFM phase transitions. On the other hand, in a nonzero
homogeneous external magnetic field , we find no evidence of a transition to
the state with spontaneous magnetization for FM interactions in three
dimensions. Nonethelesss, for AFM interactions we do get a scenario similar to
that described above for zero external magnetic field, except that the critical
temperatures are now functions of . We also find two critical field values,
, at which the reentrance phenomenon dissapears and
(), above which the PM-AFM transition temperature
vanishes.Comment: 21 pages, 6 figs. Title changed, abstract and introduction as well as
section IV were rewritten relaxing the emphasis on spin S=1 and Figs. 5 an 6
were improved in presentation. However, all the results remain valid.
Accepted for publication in Phys. Rev.