16,448 research outputs found
Relative entropy and the multi-variable multi-dimensional moment problem
Entropy-like functionals on operator algebras have been studied since the
pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most
well-known are the von Neumann entropy and a
generalization of the Kullback-Leibler distance , refered to as quantum relative entropy and used to quantify
distance between states of a quantum system. The purpose of this paper is to
explore these as regularizing functionals in seeking solutions to
multi-variable and multi-dimensional moment problems. It will be shown that
extrema can be effectively constructed via a suitable homotopy. The homotopy
approach leads naturally to a further generalization and a description of all
the solutions to such moment problems. This is accomplished by a
renormalization of a Riemannian metric induced by entropy functionals. As an
application we discuss the inverse problem of describing power spectra which
are consistent with second-order statistics, which has been the main motivation
behind the present work.Comment: 24 pages, 3 figure
Power law decay for systems of randomly coupled differential equations
We consider large random matrices with centered, independent entries but
possibly different variances. We compute the normalized trace of
for functions analytic on the spectrum of . We use these results to
compute the long time asymptotics for systems of coupled differential equations
with random coefficients. We show that when the coupling is critical the norm
squared of the solution decays like .Comment: 20 pages, Corrected a typo in Assumption (1) [after final
publication] and made other irrelevant revision
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
Mastering the Master Field
The basic concepts of non-commutative probability theory are reviewed and
applied to the large limit of matrix models. We argue that this is the
appropriate framework for constructing the master field in terms of which large
theories can be written. We explicitly construct the master field in a
number of cases including QCD. There we both give an explicit construction
of the master gauge field and construct master loop operators as well. Most
important we extend these techniques to deal with the general matrix model, in
which the matrices do not have independent distributions and are coupled. We
can thus construct the master field for any matrix model, in a well defined
Hilbert space, generated by a collection of creation and annihilation
operators---one for each matrix variable---satisfying the Cuntz algebra. We
also discuss the equations of motion obeyed by the master field.Comment: 46 pages plus 11 uuencoded eps figure
Infinite Products of Large Random Matrices and Matrix-valued Diffusion
We use an extension of the diagrammatic rules in random matrix theory to
evaluate spectral properties of finite and infinite products of large complex
matrices and large hermitian matrices. The infinite product case allows us to
define a natural matrix-valued multiplicative diffusion process. In both cases
of hermitian and complex matrices, we observe an emergence of "topological
phase transition" in the spectrum, after some critical diffusion time
is reached. In the case of the particular product of two
hermitian ensembles, we observe also an unusual localization-delocalization
phase transition in the spectrum of the considered ensemble. We verify the
analytical formulae obtained in this work by numerical simulation.Comment: 39 pages, 12 figures; v2: references added; v3: version to appear in
Nucl. Phys.
Macroscopic and microscopic (non-)universality of compact support random matrix theory
A random matrix model with a σ-model like constraint, the restricted trace ensemble (RTE), is solved in the large-n limit. In the macroscopic limit the smooth connected two-point resolvent G(z,w) is found to be non-universal, extending previous results from monomial to arbitrary polynomial potentials. Using loop equation techniques we give a closed though non-universal expression for G(z,w), which extends recursively to all higher k-point resolvents. These findings are in contrast to the usual unconstrained one-matrix model. However, in the microscopic large-n limit, which probes only correlations at distance of the mean level spacing, we are able to show that the constraint does not modify the universal sine-law. In the case of monomial potentials V(M)=M2p, we provide a relation valid for finite-n between the k-point correlation function of the RTE and the unconstrained model. In the microscopic large-n limit they coincide which proves the microscopic universality of RTEs
Quaternionic R transform and non-hermitian random matrices
Using the Cayley-Dickson construction we rephrase and review the
non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I.
Zahed, Nucl.Phys. B , 603 (1997)], that generalizes the free
probability calculus to asymptotically large non-hermitian random matrices. The
main object in this generalization is a quaternionic extension of the R
transform which is a generating function for planar (non-crossing) cumulants.
We demonstrate that the quaternionic R transform generates all connected
averages of all distinct powers of and its hermitian conjugate :
\langle\langle \frac{1}{N} \mbox{Tr} X^{a} X^{\dagger b} X^c \ldots
\rangle\rangle for . We show that the R transform for
gaussian elliptic laws is given by a simple linear quaternionic map
where
is the Cayley-Dickson pair of complex numbers forming a quaternion
. This map has five real parameters , ,
, and . We use the R transform to calculate the limiting
eigenvalue densities of several products of gaussian random matrices.Comment: 27 pages, 16 figure
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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