286,354 research outputs found
Random Matrices with Slow Correlation Decay
We consider large random matrices with a general slowly decaying correlation
among its entries. We prove universality of the local eigenvalue statistics and
optimal local laws for the resolvent away from the spectral edges, generalizing
the recent result of [arXiv:1604.08188] to allow slow correlation decay and
arbitrary expectation. The main novel tool is a systematic diagrammatic control
of a multivariate cumulant expansion.Comment: 41 pages, 1 figure. We corrected a typo in (4.1b
Statistics of Atmospheric Correlations
For a large class of quantum systems the statistical properties of their
spectrum show remarkable agreement with random matrix predictions. Recent
advances show that the scope of random matrix theory is much wider. In this
work, we show that the random matrix approach can be beneficially applied to a
completely different classical domain, namely, to the empirical correlation
matrices obtained from the analysis of the basic atmospheric parameters that
characterise the state of atmosphere. We show that the spectrum of atmospheric
correlation matrices satisfy the random matrix prescription. In particular, the
eigenmodes of the atmospheric empirical correlation matrices that have physical
significance are marked by deviations from the eigenvector distribution.Comment: 8 pages, 9 figs, revtex; To appear in Phys. Rev.
Configuration model for correlation matrices preserving the node strength
Correlation matrices are a major type of multivariate data. To examine
properties of a given correlation matrix, a common practice is to compare the
same quantity between the original correlation matrix and reference correlation
matrices, such as those derived from random matrix theory, that partially
preserve properties of the original matrix. We propose a model to generate such
reference correlation and covariance matrices for the given matrix. Correlation
matrices are often analysed as networks, which are heterogeneous across nodes
in terms of the total connectivity to other nodes for each node. Given this
background, the present algorithm generates random networks that preserve the
expectation of total connectivity of each node to other nodes, akin to
configuration models for conventional networks. Our algorithm is derived from
the maximum entropy principle. We will apply the proposed algorithm to
measurement of clustering coefficients and community detection, both of which
require a null model to assess the statistical significance of the obtained
results.Comment: 8 figures, 4 table
Noise Dressing of Financial Correlation Matrices
We show that results from the theory of random matrices are potentially of
great interest to understand the statistical structure of the empirical
correlation matrices appearing in the study of price fluctuations. The central
result of the present study is the remarkable agreement between the theoretical
prediction (based on the assumption that the correlation matrix is random) and
empirical data concerning the density of eigenvalues associated to the time
series of the different stocks of the S&P500 (or other major markets). In
particular the present study raises serious doubts on the blind use of
empirical correlation matrices for risk management.Comment: Latex (Revtex) 3 pp + 2 postscript figures (in-text
Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
We compare finite rank perturbations of the following three ensembles of
complex rectangular random matrices: First, a generalised Wishart ensemble with
one random and two fixed correlation matrices introduced by Borodin and
P\'ech\'e, second, the product of two independent random matrices where one has
correlated entries, and third, the case when the two random matrices become
also coupled through a fixed matrix. The singular value statistics of all three
ensembles is shown to be determinantal and we derive double contour integral
representations for their respective kernels. Three different kernels are found
in the limit of infinite matrix dimension at the origin of the spectrum. They
depend on finite rank perturbations of the correlation and coupling matrices
and are shown to be integrable. The first kernel (I) is found for two
independent matrices from the second, and two weakly coupled matrices from the
third ensemble. It generalises the Meijer -kernel for two independent and
uncorrelated matrices. The third kernel (III) is obtained for the generalised
Wishart ensemble and for two strongly coupled matrices. It further generalises
the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II),
found for the ensemble of two coupled matrices, provides an interpolation
between the kernels (I) and (III), generalising previous findings of part of
the authors.Comment: 39 pages, 4 figures; v2: 43 pages, presentation of Thm 1.4 improved,
alternative proof of Prop 3.1 and reference added; v3: final typo
corrections, to appear in AIHP Probabilite et Statistiqu
- …