190 research outputs found

### 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 $N_c$ 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 $d=2$ and $d=3$. Our analysis leads
us to speculate that the universal behavior observed in these lattice
simulations might be a generic feature of confinement, also in $d=4$ Yang-Mills
theory.Comment: 4 pages, no figures- Some rewriting - Typos corrected - References
completed and some correcte

### 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)

### 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

### 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 $n$-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

### 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

### Loop models, random matrices and planar algebras

We define matrix models that converge to the generating functions of a wide
variety of loop models with fugacity taken in sets with an accumulation point.
The latter can also be seen as moments of a non-commutative law on a subfactor
planar algebra. We apply this construction to compute the generating functions
of the Potts model on a random planar map

### Real symmetric random matrices and paths counting

Exact evaluation of $$ is here performed for real symmetric
matrices $S$ of arbitrary order $n$, up to some integer $p$, 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 $n$ limit. They also are a straightforward tool to examine a variety of
rescalings of the entries in the large $n$ limit.Comment: 23 pages, 10 figures, revised pape

### Adding and multiplying random matrices: a generalization of Voiculescu's formulae

In this paper, we give an elementary proof of the additivity of the
functional inverses of the resolvents of large $N$ random matrices, using
recently developed matrix model techniques. This proof also gives a very
natural generalization of these formulae to the case of measures with an
external field. A similar approach yields a relation of the same type for
multiplication of random matrices.Comment: 11 pages, harvmac. revised x 2: refs and minor comments adde

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