73 research outputs found
Random matrix approach in search for weak signals immersed in background noise
We present new, original and alternative method for searching signals coded
in noisy data. The method is based on the properties of random matrix
eigenvalue spectra. First, we describe general ideas and support them with
results of numerical simulations for basic periodic signals immersed in
artificial stochastic noise. Then, the main effort is put to examine the
strength of a new method in investigation of data content taken from the real
astrophysical NAUTILUS detector, searching for the presence of gravitational
waves. Our method discovers some previously unknown problems with data
aggregation in this experiment. We provide also the results of new method
applied to the entire respond signal from ground based detectors in future
experimental activities with reduced background noise level. We indicate good
performance of our method what makes it a positive predictor for further
applications in many areas.Comment: 15 pages, 16 figure
Number statistics for -ensembles of random matrices: applications to trapped fermions at zero temperature
Let be the probability that a
-ensemble of random matrices with confining potential
has eigenvalues inside an interval of the real
line. We introduce a general formalism, based on the Coulomb gas technique and
the resolvent method, to compute analytically for large . We show that this probability scales for large
as , where is the Dyson index of the
ensemble. The rate function , independent of ,
is computed in terms of single integrals that can be easily evaluated
numerically. The general formalism is then applied to the classical
-Gaussian (), -Wishart () and
-Cauchy () ensembles. Expanding the rate function
around its minimum, we find that generically the number variance exhibits a non-monotonic behavior as a function of the size
of the interval, with a maximum that can be precisely characterized. These
analytical results, corroborated by numerical simulations, provide the full
counting statistics of many systems where random matrix models apply. In
particular, we present results for the full counting statistics of zero
temperature one-dimensional spinless fermions in a harmonic trap.Comment: 34 pages, 19 figure
The sloppy model universality class and the Vandermonde matrix
In a variety of contexts, physicists study complex, nonlinear models with
many unknown or tunable parameters to explain experimental data. We explain why
such systems so often are sloppy; the system behavior depends only on a few
`stiff' combinations of the parameters and is unchanged as other `sloppy'
parameter combinations vary by orders of magnitude. We contrast examples of
sloppy models (from systems biology, variational quantum Monte Carlo, and
common data fitting) with systems which are not sloppy (multidimensional linear
regression, random matrix ensembles). We observe that the eigenvalue spectra
for the sensitivity of sloppy models have a striking, characteristic form, with
a density of logarithms of eigenvalues which is roughly constant over a large
range. We suggest that the common features of sloppy models indicate that they
may belong to a common universality class. In particular, we motivate focusing
on a Vandermonde ensemble of multiparameter nonlinear models and show in one
limit that they exhibit the universal features of sloppy models.Comment: New content adde
Controlling Light Through Optical Disordered Media : Transmission Matrix Approach
We experimentally measure the monochromatic transmission matrix (TM) of an
optical multiple scattering medium using a spatial light modulator together
with a phase-shifting interferometry measurement method. The TM contains all
information needed to shape the scattered output field at will or to detect an
image through the medium. We confront theory and experiment for these
applications and we study the effect of noise on the reconstruction method. We
also extracted from the TM informations about the statistical properties of the
medium and the light transport whitin it. In particular, we are able to isolate
the contributions of the Memory Effect (ME) and measure its attenuation length
Detecting entanglement of random states with an entanglement witness
The entanglement content of high-dimensional random pure states is almost
maximal, nevertheless, we show that, due to the complexity of such states, the
detection of their entanglement using witness operators is rather difficult. We
discuss the case of unknown random states, and the case of known random states
for which we can optimize the entanglement witness. Moreover, we show that
coarse graining, modeled by considering mixtures of m random states instead of
pure ones, leads to a decay in the entanglement detection probability
exponential with m. Our results also allow to explain the emergence of
classicality in coarse grained quantum chaotic dynamics.Comment: 14 pages, 4 figures; minor typos correcte
Spectra of Empirical Auto-Covariance Matrices
We compute spectra of sample auto-covariance matrices of second order
stationary stochastic processes. We look at a limit in which both the matrix
dimension and the sample size used to define empirical averages
diverge, with their ratio kept fixed. We find a remarkable scaling
relation which expresses the spectral density of sample
auto-covariance matrices for processes with dynamical correlations as a
continuous superposition of appropriately rescaled copies of the spectral
density for a sequence of uncorrelated random
variables. The rescaling factors are given by the Fourier transform
of the auto-covariance function of the stochastic process. We also obtain a
closed-form approximation for the scaling function
. This depends on the shape parameter , but
is otherwise universal: it is independent of the details of the underlying
random variables, provided only they have finite variance. Our results are
corroborated by numerical simulations using auto-regressive processes.Comment: 4 pages, 2 figure
Rates of convergence for empirical spectral measures: a soft approach
Understanding the limiting behavior of eigenvalues of random matrices is the
central problem of random matrix theory. Classical limit results are known for
many models, and there has been significant recent progress in obtaining more
quantitative, non-asymptotic results. In this paper, we describe a systematic
approach to bounding rates of convergence and proving tail inequalities for the
empirical spectral measures of a wide variety of random matrix ensembles. We
illustrate the approach by proving asymptotically almost sure rates of
convergence of the empirical spectral measure in the following ensembles:
Wigner matrices, Wishart matrices, Haar-distributed matrices from the compact
classical groups, powers of Haar matrices, randomized sums and random
compressions of Hermitian matrices, a random matrix model for the Hamiltonians
of quantum spin glasses, and finally the complex Ginibre ensemble. Many of the
results appeared previously and are being collected and described here as
illustrations of the general method; however, some details (particularly in the
Wigner and Wishart cases) are new.
Our approach makes use of techniques from probability in Banach spaces, in
particular concentration of measure and bounds for suprema of stochastic
processes, in combination with more classical tools from matrix analysis,
approximation theory, and Fourier analysis. It is highly flexible, as evidenced
by the broad list of examples. It is moreover based largely on "soft" methods,
and involves little hard analysis
Asymmetric correlation matrices: an analysis of financial data
We analyze the spectral properties of correlation matrices between distinct
statistical systems. Such matrices are intrinsically non symmetric, and lend
themselves to extend the spectral analyses usually performed on standard
Pearson correlation matrices to the realm of complex eigenvalues. We employ
some recent random matrix theory results on the average eigenvalue density of
this type of matrices to distinguish between noise and non trivial correlation
structures, and we focus on financial data as a case study. Namely, we employ
daily prices of stocks belonging to the American and British stock exchanges,
and look for the emergence of correlations between two such markets in the
eigenvalue spectrum of their non symmetric correlation matrix. We find several
non trivial results, also when considering time-lagged correlations over short
lags, and we corroborate our findings by additionally studying the asymmetric
correlation matrix of the principal components of our datasets.Comment: Revised version; 11 pages, 13 figure
Correlators for the Wigner–Smith time-delay matrix of chaotic cavities
We study the Wigner–Smith time-delay matrix Q of a ballistic quantum dot supporting N scattering channels. We compute the v-point correlators of the power traces Tr Qk for arbitrary v>1 at leading order for large N using techniques from the random matrix theory approach to quantum chromodynamics. We conjecture that the cumulants of the Tr Qkʼs are integer-valued at leading order in N and include a MATHEMATICA code that computes their generating functions recursively
Subsystem dynamics under random Hamiltonian evolution
We study time evolution of a subsystem's density matrix under unitary
evolution, generated by a sufficiently complex, say quantum chaotic,
Hamiltonian, modeled by a random matrix. We exactly calculate all coherences,
purity and fluctuations. We show that the reduced density matrix can be
described in terms of a noncentral correlated Wishart ensemble for which we are
able to perform analytical calculations of the eigenvalue density. Our
description accounts for a transition from an arbitrary initial state towards a
random state at large times, enabling us to determine the convergence time
after which random states are reached. We identify and describe a number of
other interesting features, like a series of collisions between the largest
eigenvalue and the bulk, accompanied by a phase transition in its distribution
function.Comment: 16 pages, 8 figures; v3: slightly re-structured and an additional
appendi
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