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

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

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

Signal from noise retrieval from one and two-point Green's function - comparison

We compare two methods of eigen-inference from large sets of data, based on the analysis of one-point and two-point Green's functions, respectively. Our analysis points at the superiority of eigen-inference based on one-point Green's function. First, the applied by us method based on Pad?e approximants is orders of magnitude faster comparing to the eigen-inference based on uctuations (two-point Green's functions). Second, we have identified the source of potential instability of the two-point Green's function method, as arising from the spurious zero and negative modes of the estimator for a variance operator of the certain multidimensional Gaussian distribution, inherent for the two-point Green's function eigen-inference method. Third, we have presented the cases of eigen-inference based on negative spectral moments, for strictly positive spectra. Finally, we have compared the cases of eigen-inference of real-valued and complex-valued correlated Wishart distributions, reinforcing our conclusions on an advantage of the one-point Green's function method.Comment: 14 pages, 8 figures, 3 table

Generating random density matrices

We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi--partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N \to \infty, by the Marchenko-Pastur distribution.Comment: 13 pages in latex with 8 figures include

Summing free unitary random matrices

I use quaternion free probability calculus - an extension of free probability to non-Hermitian matrices (which is introduced in a succinct but self-contained way) - to derive in the large-size limit the mean densities of the eigenvalues and singular values of sums of independent unitary random matrices, weighted by complex numbers. In the case of CUE summands, I write them in terms of two "master equations," which I then solve and numerically test in four specific cases. I conjecture a finite-size extension of these results, exploiting the complementary error function. I prove a central limit theorem, and its first sub-leading correction, for independent identically-distributed zero-drift unitary random matrices.Comment: 17 pages, 15 figure