153 research outputs found
On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry
We suggest a method of studying the joint probability density (JPD) of an
eigenvalue and the associated 'non-orthogonality overlap factor' (also known as
the 'eigenvalue condition number') of the left and right eigenvectors for
non-selfadjoint Gaussian random matrices of size . First we derive
the general finite expression for the JPD of a real eigenvalue
and the associated non-orthogonality factor in the real Ginibre ensemble, and
then analyze its 'bulk' and 'edge' scaling limits. The ensuing distribution is
maximally heavy-tailed, so that all integer moments beyond normalization are
divergent. A similar calculation for a complex eigenvalue and the
associated non-orthogonality factor in the complex Ginibre ensemble is
presented as well and yields a distribution with the finite first moment. Its
'bulk' scaling limit yields a distribution whose first moment reproduces the
well-known result of Chalker and Mehlig \cite{ChalkerMehlig1998}, and we
provide the 'edge' scaling distribution for this case as well. Our method
involves evaluating the ensemble average of products and ratios of integer and
half-integer powers of characteristic polynomials for Ginibre matrices, which
we perform in the framework of a supersymmetry approach. Our paper complements
recent studies by Bourgade and Dubach \cite{BourgadeDubach}.Comment: published versio
Spectra of Random Matrices Close to Unitary and Scattering Theory for Discrete-Time Systems
We analyze statistical properties of complex eigenvalues of random matrices
close to unitary. Such matrices appear naturally when considering
quantized chaotic maps within a general theory of open linear stationary
systems with discrete time. Deviation from unitarity are characterized by rank
and eigenvalues of the matrix . For the case M=1 we solve the problem completely
by deriving the joint probability density of eigenvalues and calculating all
point correlation functions. For a general case we present the correlation
function of secular determinants.Comment: 4 pages, latex, no figures, a few misprints are correcte
High-Dimensional Random Fields and Random Matrix Theory
Our goal is to discuss in detail the calculation of the mean number of
stationary points and minima for random isotropic Gaussian fields on a sphere
as well as for stationary Gaussian random fields in a background parabolic
confinement. After developing the general formalism based on the
high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory
(RMT) techniques to perform analysis of the random energy landscape of spin
spherical spinglasses and a related glass model, both displaying a
zero-temperature one-step replica symmetry breaking glass transition as a
function of control parameters (e.g. a magnetic field or curvature of the
confining potential). A particular emphasis of the presented analysis is on
understanding in detail the picture of "topology trivialization" (in the sense
of drastic reduction of the number of stationary points) of the landscape which
takes place in the vicinity of the zero-temperature glass transition in both
models. We will reveal the important role of the GOE "edge scaling" spectral
region and the Tracy-Widom distribution of the maximal eigenvalue of GOE
matrices for providing an accurate quantitative description of the universal
features of the topology trivialization scenario.Comment: 40 pages; 2 figures; In this version the original lecture notes are
converted to an article format, new Eqs. (82)-(85) and Appendix about
anisotropic fields added, noticed misprints corrected, references updated.
references update
A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise
An encryption of a signal is a random mapping which can be corrupted
by an additive noise. Given the Encryption Redundancy Parameter (ERP)
, the signal strength parameter , and
the ('bare') noise-to-signal ratio (NSR) , we consider the problem
of reconstructing from its corrupted image by a Least Square Scheme
for a certain class of random Gaussian mappings. The problem is equivalent to
finding the configuration of minimal energy in a certain version of spherical
spin glass model, with squared Gaussian-distributed random potential. We use
the Parisi replica symmetry breaking scheme to evaluate the mean overlap
between the original signal and its recovered image
(known as 'estimator') as , which is a measure of the quality of
the signal reconstruction. We explicitly analyze the general case of
linear-quadratic family of random mappings and discuss the full curve. When nonlinearity exceeds a certain threshold but redundancy
is not yet too big, the replica symmetric solution is necessarily broken in
some interval of NSR. We show that encryptions with a nonvanishing linear
component permit reconstructions with for any and any
, with as . In
contrast, for the case of purely quadratic nonlinearity, for any ERP
there exists a threshold NSR value such that for
making the reconstruction impossible. The behaviour
close to the threshold is given by and
is controlled by the replica symmetry breaking mechanism.Comment: 33 pages, 5 figure
The Spectral Autocorrelation Function in Weakly Open Chaotic Systems: Indirect Photodissociation of Molecules
We derive the statistical limit of the spectral autocorrelation function and
of the survival probability for the indirect photodissociation of molecules in
the regime of non-overlapping resonances. The results are derived in the
framework of random matrix theory, and hold more generally for any chaotic
quantum system that is weakly coupled to the continuum. The "correlation hole"
that characterizes the spectral autocorrelation in the bound molecule
diminishes as the typical average total width of a resonance increases.Comment: 13 pages, 1 Postscript figure included, RevTe
On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials
Correlation functions involving products and ratios of half-integer powers of
characteristic polynomials of random matrices from the Gaussian Orthogonal
Ensemble (GOE) frequently arise in applications of Random Matrix Theory (RMT)
to physics of quantum chaotic systems, and beyond. We provide an explicit
evaluation of the large- limits of a few non-trivial objects of that sort
within a variant of the supersymmetry formalism, and via a related but
different method. As one of the applications we derive the distribution of an
off-diagonal entry of the resolvent (or Wigner -matrix) of GOE
matrices which, among other things, is of relevance for experiments on chaotic
wave scattering in electromagnetic resonators.Comment: 25 pages (2 figures); published version (conclusion added, minor
changes
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