419 research outputs found
Hybrid normed ideal perturbations of n-tuples of operators II: weak wave operators
We prove a general weak existence theorem for wave operators for hybrid
normed ideal perturbations. We then use this result to prove the invariance of
Lebesgue absolutely continuous parts of n-tuples of commuting hermitian
operators under hybrid normed ideal perturbations from a class studied in the
first paper of this series.Comment: 9 page
Volumes of Restricted Minkowski Sums and the Free Analogue of the Entropy Power Inequality
In noncommutative probability theory independence can be based on free
products instead of tensor products. This yields a highly noncommutative
theory: free probability . Here we show that the classical Shannon's entropy
power inequality has a counterpart for the free analogue of entropy .
The free entropy (introduced recently by the second named author),
consistently with Boltzmann's formula , was defined via volumes of
matricial microstates. Proving the free entropy power inequality naturally
becomes a geometric question.
Restricting the Minkowski sum of two sets means to specify the set of pairs
of points which will be added. The relevant inequality, which holds when the
set of "addable" points is sufficiently large, differs from the Brunn-Minkowski
inequality by having the exponent replaced by . Its proof uses the
rearrangement inequality of Brascamp-Lieb-L\"uttinger
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.
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
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
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
Uniform version of Weyl-von Neumann theorem
We prove a "quantified" version of the Weyl-von Neumann theorem, more
precisely, we estimate the ranks of approximants to compact operators appearing
in the Voiculescu's theorem applied to commutative algebras. This allows
considerable simplifications in uniform K-homology theory, namely it shows that
one can represent all the uniform K-homology classes on a fixed Hilbert space
with a fixed *-representation of C_0(X), for a large class of spaces X
Asymptotic mean density of sub-unitary ensemble
The large N limit of mean spectral density for the ensemble of NxN
sub-unitary matrices derived by Wei and Fyodorov (J. Phys. A: Math. Theor. 41
(2008) 50201) is calculated by a modification of the saddle point method. It is
shown that the result coincides with the one obtained within the free
probability theory by Haagerup and Larsen (J. Funct. Anal. 176 (2000) 331)
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
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