22 research outputs found
Invariant sums of random matrices and the onset of level repulsion
We compute analytically the joint probability density of eigenvalues and the
level spacing statistics for an ensemble of random matrices with interesting
features. It is invariant under the standard symmetry groups (orthogonal and
unitary) and yet the interaction between eigenvalues is not Vandermondian. The
ensemble contains real symmetric or complex hermitian matrices of
the form or respectively. The
diagonal matrices
are
constructed from real eigenvalues drawn \emph{independently} from distributions
, while the matrices and are all
orthogonal or unitary. The average is simultaneously
performed over the symmetry group and the joint distribution of
. We focus on the limits i.) and ii.)
, with . In the limit i.), the resulting sum
develops level repulsion even though the original matrices do not feature it,
and classical RMT universality is restored asymptotically. In the limit ii.)
the spacing distribution attains scaling forms that are computed exactly: for
the orthogonal case, we recover the Wigner's surmise, while for the
unitary case an entirely new universal distribution is obtained. Our results
allow to probe analytically the microscopic statistics of the sum of random
matrices that become asymptotically free. We also give an interpretation of
this model in terms of radial random walks in a matrix space. The analytical
results are corroborated by numerical simulations.Comment: 19 pag., 6 fig. - published versio
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
Statistical mechanics of random graphs
We discuss various aspects of the statistical formulation of the theory of
random graphs, with emphasis on results obtained in a series of our recent
publications.Comment: 6 pages (Talk at Conference: Applications of Physics in Financial
Analysis 4, Warsaw 13-15 Nov. 2003
The Ising model on a random lattice with a coordnation numer equal 3
The micro- and grand-canonical partition functions for a system of spins on a dynamical two-dimensional random spherical surface with a coordination number 3 restricted to the set of lattices without the ‘tadpole’ and ‘self-energy’ insertions is calculated. The critical properties are shown to be the same as in the case of the unrestricted set of the lattices
Signal and Noise in Financial Correlation Matrices
Using Random Matrix Theory one can derive exact relations between the
eigenvalue spectrum of the covariance matrix and the eigenvalue spectrum of its
estimator (experimentally measured correlation matrix). These relations will be
used to analyze a particular case of the correlations in financial series and
to show that contrary to earlier claims, correlations can be measured also in
the ``random'' part of the spectrum. Implications for the portfolio
optimization are briefly discussed.Comment: 6 pages + 2 figures, corrected references, Talk at Conference:
Applications of Physics in Financial Analysis 4, Warsaw, 13-15 November 200
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 (Z. Burda, A. Görlich, J. Jurkiewicz, B. Waclaw, cond-mat/0508341). The method allows for exact determination of the experimental spectrum for a given covariance matrix and given correlations between samples in the limit 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
Dysonian dynamics of the Ginibre ensemble
We study the time evolution of Ginibre matrices whose elements undergo
Brownian motion. The non-Hermitian character of the Ginibre ensemble binds the
dynamics of eigenvalues to the evolution of eigenvectors in a non-trivial way,
leading to a system of coupled nonlinear equations resembling those for
turbulent systems. We formulate a mathematical framework allowing simultaneous
description of the flow of eigenvalues and eigenvectors, and we unravel a
hidden dynamics as a function of new complex variable, which in the standard
description is treated as a regulator only. We solve the evolution equations
for large matrices and demonstrate that the non-analytic behavior of the
Green's functions is associated with a shock wave stemming from a Burgers-like
equation describing correlations of eigenvectors. We conjecture that the hidden
dynamics, that we observe for the Ginibre ensemble, is a general feature of
non-Hermitian random matrix models and is relevant to related physical
applications.Comment: 5 pages, 2 figure
A Random Matrix Approach to VARMA Processes
We apply random matrix theory to derive spectral density of large sample
covariance matrices generated by multivariate VMA(q), VAR(q) and VARMA(q1,q2)
processes. In particular, we consider a limit where the number of random
variables N and the number of consecutive time measurements T are large but the
ratio N/T is fixed. In this regime the underlying random matrices are
asymptotically equivalent to Free Random Variables (FRV). We apply the FRV
calculus to calculate the eigenvalue density of the sample covariance for
several VARMA-type processes. We explicitly solve the VARMA(1,1) case and
demonstrate a perfect agreement between the analytical result and the spectra
obtained by Monte Carlo simulations. The proposed method is purely algebraic
and can be easily generalized to q1>1 and q2>1.Comment: 16 pages, 6 figures, submitted to New Journal of Physic
Free Random Levy Variables and Financial Probabilities
We suggest that Free Random Variables, represented here by large random
matrices with spectral Levy disorder, may be relevant for several problems
related to the modeling of financial systems. In particular, we consider a
financial covariance matrix composed of asymmetric and free random Levy
matrices. We derive an algebraic equation for the resolvent and solve it to
extract the spectral density. The free eigenvalue spectrum is in remarkable
agreement with the one obtained from the covariance matrix of the SP500
financial market.Comment: 8 pages with 2 EPS figures; talk given by M.A. Nowak at NATO Advanced
Research Workshop ``Applications of Physics to Economic Modeling'', Prague,
8-10 February, 200
Wealth condensation and "corruption" in a toy model
We discuss the wealth condensation mechanism in a simple toy economy in which individual agent’s wealths are distributed according to a Pareto power law and the overall wealth is fixed. The observed behaviour is the manifestation of a transition which occurs in Zero Range Processes (ZRPs) or "balls in boxes" models. An amusing feature of the transition in this context is that the condensation can be induced by increasing the exponent in the power law, which one might have naively assumed penalised greater wealths more