1,098 research outputs found
Spectral properties of empirical covariance matrices for data with power-law tails
We present an analytic method for calculating spectral densities of empirical
covariance matrices for correlated data. In this approach the data is
represented as a rectangular random matrix whose columns correspond to sampled
states of the system. The method is applicable to a class of random matrices
with radial measures including those with heavy (power-law) tails in the
probability distribution. As an example we apply it to a multivariate Student
distribution.Comment: 9 pages, 3 figures, references adde
Collapse of 4D random geometries
We extend the analysis of the Backgammon model to an ensemble with a fixed
number of balls and a fluctuating number of boxes. In this ensemble the model
exhibits a first order phase transition analogous to the one in higher
dimensional simplicial gravity. The transition relies on a kinematic
condensation and reflects a crisis of the integration measure which is probably
a part of the more general problem with the measure for functional integration
over higher (d>2) dimensional Riemannian structures.Comment: 7 pages, Latex2e, 2 figures (.eps
Counting metastable states of Ising spin glasses on arbitrary graphs
Using a field-theoretical representation of the Tanaka-Edwards integral we
develop a method to systematically compute the number N_s of 1-spin-stable
states (local energy minima) of a glassy Ising system with nearest-neighbor
interactions and random Gaussian couplings on an arbitrary graph. In
particular, we use this method to determine N_s for K-regular random graphs and
d-dimensional regular lattices for d=2,3. The method works also for other
graphs. Excellent accuracy of the results allows us to observe that the number
of local energy minima depends mainly on local properties of the graph on which
the spin glass is defined.Comment: 8 pages, 4 figures (one in color), additional materials can be found
under http://www.physik.uni-leipzig.de/~waclaw/glasses-data.ht
Commutative law for products of infinitely large isotropic random matrices
Ensembles of isotropic random matrices are defined by the invariance of the
probability measure under the left (and right) multiplication by an arbitrary
unitary matrix. We show that the multiplication of large isotropic random
matrices is spectrally commutative and self-averaging in the limit of infinite
matrix size . The notion of spectral commutativity means
that the eigenvalue density of a product ABC... of such matrices is independent
of the order of matrix multiplication, for example the matrix ABCD has the same
eigenvalue density as ADCB. In turn, the notion of self-averaging means that
the product of n independent but identically distributed random matrices, which
we symbolically denote by AAA..., has the same eigenvalue density as the
corresponding power A^n of a single matrix drawn from the underlying matrix
ensemble. For example, the eigenvalue density of ABCCABC is the same as of
A^2B^2C^3. We also discuss the singular behavior of the eigenvalue and singular
value densities of isotropic matrices and their products for small eigenvalues
. We show that the singularities at the origin of the
eigenvalue density and of the singular value density are in one-to-one
correspondence in the limit : the eigenvalue density of
an isotropic random matrix has a power law singularity at the origin with a power when and only when the density of
its singular values has a power law singularity with a
power . These results are obtained analytically in the limit
. We supplement these results with numerical simulations
for large but finite N and discuss finite size effects for the most common
ensembles of isotropic random matrices.Comment: 15 pages, 4 figure
Correlation functions and critical behaviour on fluctuating geometries
We study the two-point correlation function in the model of branched polymers
and its relation to the critical behaviour of the model. We show that the
correlation function has a universal scaling form in the generic phase with the
only scale given by the size of the polymer. We show that the origin of the
singularity of the free energy at the critical point is different from that in
the standard statistical models. The transition is related to the change of the
dimensionality of the system.Comment: 10 Pages, Latex2e, uses elsart.cls, 1 figure include
Phase diagram of the mean field model of simplicial gravity
We discuss the phase diagram of the balls in boxes model, with a varying
number of boxes. The model can be regarded as a mean-field model of simplicial
gravity. We analyse in detail the case of weights of the form , which correspond to the measure term introduced in the simplicial
quantum gravity simulations. The system has two phases~: {\em elongated} ({\em
fluid}) and {\em crumpled}. For the transition between
these two phases is first order, while for it is continuous.
The transition becomes softer when approaches unity and eventually
disappears at . We then generalise the discussion to an arbitrary set
of weights. Finally, we show that if one introduces an additional kinematic
bound on the average density of balls per box then a new {\em condensed} phase
appears in the phase diagram. It bears some similarity to the {\em crinkled}
phase of simplicial gravity discussed recently in models of gravity interacting
with matter fields.Comment: 15 pages, 5 figure
Eigenvalue density of empirical covariance matrix for correlated samples
We describe a method to determine the eigenvalue density of empirical
covariance matrix in the presence of correlations between samples. This is a
straightforward generalization of the method developed earlier by the authors
for uncorrelated samples. The method allows for exact determination of the
experimental spectrum for a given covariance matrix and given correlations
between samples in the limit of large N and N/T=r=const with N being the number
of degrees of freedom and T being the number of samples. We discuss the effect
of correlations on several examples.Comment: 12 pages, 5 figures, to appear in Acta Phys. Pol. B (Proceedings of
the conference on `Applications of Random Matrix Theory to Economy and Other
Complex Systems', May 25-28, 2005, Cracow, Polan
- …