88 research outputs found
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
Random matrix model for QCD_3 staggered fermions
We show that the lowest part of the eigenvalue density of the staggered
fermion operator in lattice QCD_3 at small lattice coupling constant beta has
exactly the same shape as in QCD_4. This observation is quite surprising, since
universal properties of the QCD_3 Dirac operator are expected to be described
by a non-chiral matrix model. We show that this effect is related to the
specific nature of the staggered fermion discretization and that the eigenvalue
density evolves towards the non-chiral random matrix prediction when beta is
increased and the continuum limit is approached. We propose a two-matrix model
with one free parameter which interpolates between the two limits and very well
mimics the pattern of evolution with beta of the eigenvalue density of the
staggered fermion operator in QCD_3.Comment: 8 pages 4 figure
Spectra of sparse non-Hermitian random matrices: an analytical solution
We present the exact analytical expression for the spectrum of a sparse
non-Hermitian random matrix ensemble, generalizing two classical results in
random-matrix theory: this analytical expression forms a non-Hermitian version
of the Kesten-Mckay law as well as a sparse realization of Girko's elliptic
law. Our exact result opens new perspectives in the study of several physical
problems modelled on sparse random graphs. In this context, we show
analytically that the convergence rate of a transport process on a very sparse
graph depends upon the degree of symmetry of the edges in a non-monotonous way.Comment: 5 pages, 5 figures, 12 pages supplemental materia
Chiral Symmetry Breaking and the Dirac Spectrum at Nonzero Chemical Potential
The relation between the spectral density of the QCD Dirac operator at
nonzero baryon chemical potential and the chiral condensate is investigated. We
use the analytical result for the eigenvalue density in the microscopic regime
which shows oscillations with a period that scales as 1/V and an amplitude that
diverges exponentially with the volume . We find that the discontinuity
of the chiral condensate is due to the whole oscillating region rather than to
an accumulation of eigenvalues at the origin. These results also extend beyond
the microscopic regime to chemical potentials .Comment: 4 pages, 1 figur
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
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
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
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
Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition
6 pages, 2 figures. Final version6 pages, 2 figures. Final version6 pages, 2 figures. Final versionWe study the distribution of the mean radial displacement of charges of a 2D one-component plasma in the thermodynamic limit at finite temperature . We compute explicitly the large deviation functions showing the emergence of a fourth-order phase transition as a consequence of a change of topology in the plasma distribution. This weak phase transition occurs exactly at the ground state of the plasma. These results have been compared with the integrable case (finite ) of plasma parameter . In this case the problem can be mapped to the stationary properties of 2D Dyson Brownian particles and to a non-Hermitian matrix model
Large scale correlations in normal and general non-Hermitian matrix ensembles
We compute the large scale (macroscopic) correlations in ensembles of normal
random matrices with an arbitrary measure and in ensembles of general
non-Hermition matrices with a class of non-Gaussian measures. In both cases the
eigenvalues are complex and in the large limit they occupy a domain in the
complex plane. For the case when the support of eigenvalues is a connected
compact domain, we compute two-, three- and four-point connected correlation
functions in the first non-vanishing order in 1/N in a manner that the
algorithm of computing higher correlations becomes clear. The correlation
functions are expressed through the solution of the Dirichlet boundary problem
in the domain complementary to the support of eigenvalues. The two-point
correlation functions are shown to be universal in the sense that they depend
only on the support of eigenvalues and are expressed through the Dirichlet
Green function of its complement.Comment: 16 pages, 1 figure, LaTeX, submitted to J. Phys. A special issue on
random matrices, minor corrections, references adde
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