470 research outputs found
Second Order Freeness and Fluctuations of Random Matrices: II. Unitary Random Matrices
We extend the relation between random matrices and free probability theory
from the level of expectations to the level of fluctuations. We show how the
concept of "second order freeness", which was introduced in Part I, allows one
to understand global fluctuations of Haar distributed unitary random matrices.
In particular, independence between the unitary ensemble and another ensemble
goes in the large limit over into asymptotic second order freeness. Two
important consequences of our general theory are: (i) we obtain a natural
generalization of a theorem of Diaconis and Shahshahani to the case of several
independent unitary matrices; (ii) we can show that global fluctuations in
unitarily invariant multi-matrix models are not universal.Comment: 31 pages, new section on failure of universality added, typos
corrected, additional explanation
Fuzzy spaces and new random matrix ensembles
We analyze the expectation value of observables in a scalar theory on the
fuzzy two sphere, represented as a generalized hermitian matrix model. We
calculate explicitly the form of the expectation values in the large-N limit
and demonstrate that, for any single kind of field (matrix), the distribution
of its eigenvalues is still a Wigner semicircle but with a renormalized radius.
For observables involving more than one type of matrix we obtain a new
distribution corresponding to correlated Wigner semicircles.Comment: 12 pages, 1 figure; version to appear in Phys. Rev.
Spectrum of the Product of Independent Random Gaussian Matrices
We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M
independent NxN Gaussian random matrices in the large-N limit is rotationally
symmetric in the complex plane and is given by a simple expression
rho(z,\bar{z}) = 1/(M\pi\sigma^2} |z|^{-2+2/M} for |z|<\sigma, and is zero for
|z|> \sigma. The parameter \sigma corresponds to the radius of the circular
support and is related to the amplitude of the Gaussian fluctuations. This form
of the eigenvalue density is highly universal. It is identical for products of
Gaussian Hermitian, non-Hermitian, real or complex random matrices. It does not
change even if the matrices in the product are taken from different Gaussian
ensembles. We present a self-contained derivation of this result using a planar
diagrammatic technique for Gaussian matrices. We also give a numerical evidence
suggesting that this result applies also to matrices whose elements are
independent, centered random variables with a finite variance.Comment: 16 pages, 6 figures, minor changes, some references adde
Large N_c confinement and turbulence
We suggest that the transition that occurs at large in the eigenvalue
distribution of a Wilson loop may have a turbulent origin. We arrived at this
conclusion by studying the complex-valued inviscid Burgers-Hopf equation that
corresponds to the Makeenko-Migdal loop equation, and we demonstrate the
appearance of a shock in the spectral flow of the Wilson loop eigenvalues. This
picture supplements that of the Durhuus-Olesen transition with a particular
realization of disorder. The critical behavior at the formation of the shock
allows us to infer exponents that have been measured recently in lattice
simulations by Narayanan and Neuberger in and . Our analysis leads
us to speculate that the universal behavior observed in these lattice
simulations might be a generic feature of confinement, also in Yang-Mills
theory.Comment: 4 pages, no figures- Some rewriting - Typos corrected - References
completed and some correcte
Free Random Levy Variables and Financial Probabilities
We suggest that Free Random Variables, represented here by large random
matrices with spectral Levy disorder, may be relevant for several problems
related to the modeling of financial systems. In particular, we consider a
financial covariance matrix composed of asymmetric and free random Levy
matrices. We derive an algebraic equation for the resolvent and solve it to
extract the spectral density. The free eigenvalue spectrum is in remarkable
agreement with the one obtained from the covariance matrix of the SP500
financial market.Comment: 8 pages with 2 EPS figures; talk given by M.A. Nowak at NATO Advanced
Research Workshop ``Applications of Physics to Economic Modeling'', Prague,
8-10 February, 200
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
IMF effect on sporadic-E layers at two northern polar cap sites: Part I ? Statistical study
International audienceIn this paper we investigate the relationship between polar cap sporadic-E layers and the direction of the interplanetary magnetic field (IMF) using a 2-year database from Longyearbyen (75.2 CGM Lat, Svalbard) and Thule (85.4 CGM Lat, Greenland). It is found that the MLT distributions of sporadic-E occurrence are different at the two stations, but both are related to the IMF orientation. This relationship, however, changes from the centre of the polar cap to its border. Layers are more frequent during positive By at both stations. This effect is particularly strong in the central polar cap at Thule, where a weak effect associated with Bz is also observed, with positive Bz correlating with a higher occurrence of Es. Close to the polar cap boundary, at Longyearbyen, the By effect is weaker than at Thule. On the other hand, Bz plays there an equally important role as By, with negative Bz correlating with the Es occurrence. Since Es layers can be created by electric fields at high latitudes, a possible explanation for the observations is that the layers are produced by the polar cap electric field controlled by the IMF. Using electric field estimates calculated by means of the statistical APL convection model from IMF observations, we find that the diurnal distributions of sporadic-E occurrence can generally be explained in terms of the electric field mechanism. However, other factors must be considered to explain why more layers occur during positive than during negative By and why the Bz dependence of layer occurrence in the central polar cap is different from that at the polar cap boundary
Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices
We derive exact analytic expressions for the distributions of eigenvalues and
singular values for the product of an arbitrary number of independent
rectangular Gaussian random matrices in the limit of large matrix dimensions.
We show that they both have power-law behavior at zero and determine the
corresponding powers. We also propose a heuristic form of finite size
corrections to these expressions which very well approximates the distributions
for matrices of finite dimensions.Comment: 13 pages, 3 figure
Rigorous mean field model for CPA: Anderson model with free random variables
A model of a randomly disordered system with site-diagonal random energy
fluctuations is introduced. It is an extension of Wegner's -orbital model to
arbitrary eigenvalue distribution in the electronic level space. The new
feature is that the random energy values are not assumed to be independent at
different sites but free. Freeness of random variables is an analogue of the
concept of independence for non-commuting random operators. A possible
realization is the ensemble of at different lattice-sites randomly rotated
matrices. The one- and two-particle Green functions of the proposed hamiltonian
are calculated exactly. The eigenstates are extended and the conductivity is
nonvanishing everywhere inside the band. The long-range behaviour and the
zero-frequency limit of the two-particle Green function are universal with
respect to the eigenvalue distribution in the electronic level space. The
solutions solve the CPA-equation for the one- and two-particle Green function
of the corresponding Anderson model. Thus our (multi-site) model is a rigorous
mean field model for the (single-site) CPA. We show how the Llyod model is
included in our model and treat various kinds of noises.Comment: 24 pages, 2 diagrams, Rev-Tex. Diagrams are available from the
authors upon reques
Multiplication law and S transform for non-hermitian random matrices
We derive a multiplication law for free non-hermitian random matrices
allowing for an easy reconstruction of the two-dimensional eigenvalue
distribution of the product ensemble from the characteristics of the individual
ensembles. We define the corresponding non-hermitian S transform being a
natural generalization of the Voiculescu S transform. In addition we extend the
classical hermitian S transform approach to deal with the situation when the
random matrix ensemble factors have vanishing mean including the case when both
of them are centered. We use planar diagrammatic techniques to derive these
results.Comment: 25 pages + 11 figure
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