1,607 research outputs found
Universality and chaos for tensor products of operators
AbstractWe give sufficient conditions for the universality of tensor products {Tn⊗Rn:n∈N} of sequences of operators defined on Fréchet spaces. In particular we study when the tensor product T⊗R of two operators is chaotic in the sense of Devaney. Applications are given for natural operators on function spaces of several variables, in Infinite Holomorphy, and for multiplication operators on the algebra L(E) following the study of Kit Chan
Einstein gravity 3-point functions from conformal field theory
We study stress tensor correlation functions in four-dimensional conformal
field theories with large and a sparse spectrum. Theories in this class are
expected to have local holographic duals, so effective field theory in anti-de
Sitter suggests that the stress tensor sector should exhibit universal,
gravity-like behavior. At the linearized level, the hallmark of locality in the
emergent geometry is that stress tensor three-point functions , normally specified by three constants, should approach a universal
structure controlled by a single parameter as the gap to higher spin operators
is increased. We demonstrate this phenomenon by a direct CFT calculation.
Stress tensor exchange, by itself, violates causality and unitarity unless the
three-point functions are carefully tuned, and the unique consistent choice
exactly matches the prediction of Einstein gravity. Under some assumptions
about the other potential contributions, we conclude that this structure is
universal, and in particular, that the anomaly coefficients satisfy as conjectured by Camanho et al. The argument is based on causality of a
four-point function, with kinematics designed to probe bulk locality, and
invokes the chaos bound of Maldacena, Shenker, and Stanford.Comment: 24+9 pages; minor changes, conclusions unchange
Knotted Strange Attractors and Matrix Lorenz Systems
A generalization of the Lorenz equations is proposed where the variables take
values in a Lie algebra. The finite dimensionality of the representation
encodes the quantum fluctuations, while the non-linear nature of the equations
can describe chaotic fluctuations. We identify a criterion, for the appearance
of such non-linear terms. This depends on whether an invariant, symmetric
tensor of the algebra can vanish or not. This proposal is studied in detail for
the fundamental representation of . We find a knotted
structure for the attractor, a bimodal distribution for the largest Lyapunov
exponent and that the dynamics takes place within the Cartan subalgebra, that
does not contain only the identity matrix, thereby can describe the quantum
fluctuations.Comment: 10 pages Revtex, 3 figure
Wigner chaos and the fourth moment
We prove that a normalized sequence of multiple Wigner integrals (in a fixed
order of free Wigner chaos) converges in law to the standard semicircular
distribution if and only if the corresponding sequence of fourth moments
converges to 2, the fourth moment of the semicircular law. This extends to the
free probabilistic, setting some recent results by Nualart and Peccati on
characterizations of central limit theorems in a fixed order of Gaussian Wiener
chaos. Our proof is combinatorial, analyzing the relevant noncrossing
partitions that control the moments of the integrals. We can also use these
techniques to distinguish the first order of chaos from all others in terms of
distributions; we then use tools from the free Malliavin calculus to give
quantitative bounds on a distance between different orders of chaos. When
applied to highly symmetric kernels, our results yield a new transfer
principle, connecting central limit theorems in free Wigner chaos to those in
Gaussian Wiener chaos. We use this to prove a new free version of an important
classical theorem, the Breuer-Major theorem.Comment: Published in at http://dx.doi.org/10.1214/11-AOP657 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Quantum Magnets and Matrix Lorenz Systems
The Landau--Lifshitz--Gilbert equations for the evolution of the
magnetization, in presence of an external torque, can be cast in the form of
the Lorenz equations and, thus, can describe chaotic fluctuations. To study
quantum effects, we describe the magnetization by matrices, that take values in
a Lie algebra. The finite dimensionality of the representation encodes the
quantum fluctuations, while the non-linear nature of the equations can describe
chaotic fluctuations. We identify a criterion, for the appearance of such
non-linear terms. This depends on whether an invariant, symmetric tensor of the
algebra can vanish or not. This proposal is studied in detail for the
fundamental representation of
. We find a knotted
structure for the attractor, a bimodal distribution for the largest Lyapunov
exponent and that the dynamics takes place within the Cartan subalgebra, that
does not contain only the identity matrix, thereby can describe the quantum
fluctuations.Comment: 5 pages, 3 figures. Uses jpconf style. Presented at the ICM-SQUARE 4
conference, Madrid, August 2014. The topic is a special case of the content
of 1404.7774, currently under revisio
Tensor products of recurrent hypercyclic semigroups
We study tensor products of strongly continuous semigroups on Banach spaces
that satisfy the hypercyclicity criterion, the recurrent hypercyclicity
criterion or are chaotic.Comment: 6 pages. Final version to appear in J. Math. Anal. App
Can chaotic quantum energy levels statistics be characterized using information geometry and inference methods?
In this paper, we review our novel information geometrodynamical approach to
chaos (IGAC) on curved statistical manifolds and we emphasize the usefulness of
our information-geometrodynamical entropy (IGE) as an indicator of chaoticity
in a simple application. Furthermore, knowing that integrable and chaotic
quantum antiferromagnetic Ising chains are characterized by asymptotic
logarithmic and linear growths of their operator space entanglement entropies,
respectively, we apply our IGAC to present an alternative characterization of
such systems. Remarkably, we show that in the former case the IGE exhibits
asymptotic logarithmic growth while in the latter case the IGE exhibits
asymptotic linear growth. At this stage of its development, IGAC remains an
ambitious unifying information-geometric theoretical construct for the study of
chaotic dynamics with several unsolved problems. However, based on our recent
findings, we believe it could provide an interesting, innovative and
potentially powerful way to study and understand the very important and
challenging problems of classical and quantum chaos.Comment: 21 page
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