11,307 research outputs found
A Divergence Formula for Randomness and Dimension
If is an infinite sequence over a finite alphabet and is
a probability measure on , then the {\it dimension} of with
respect to , written , is a constructive version of
Billingsley dimension that coincides with the (constructive Hausdorff)
dimension when is the uniform probability measure. This paper
shows that and its dual \Dim^\beta(S), the {\it strong
dimension} of with respect to , can be used in conjunction with
randomness to measure the similarity of two probability measures and
on . Specifically, we prove that the {\it divergence formula}
\dim^\beta(R) = \Dim^\beta(R) =\frac{\CH(\alpha)}{\CH(\alpha) + \D(\alpha ||
\beta)} holds whenever and are computable, positive
probability measures on and is random with
respect to . In this formula, \CH(\alpha) is the Shannon entropy of
, and \D(\alpha||\beta) is the Kullback-Leibler divergence between
and . We also show that the above formula holds for all
sequences that are -normal (in the sense of Borel) when
and \Dim^\beta(R) are replaced by the more effective
finite-state dimensions \dimfs^\beta(R) and \Dimfs^\beta(R). In the course
of proving this, we also prove finite-state compression characterizations of
\dimfs^\beta(S) and \Dimfs^\beta(S).Comment: 18 page
A Divergence Formula for Randomness and Dimension (Short Version)
If is an infinite sequence over a finite alphabet and is
a probability measure on , then the {\it dimension} of with
respect to , written , is a constructive version of
Billingsley dimension that coincides with the (constructive Hausdorff)
dimension when is the uniform probability measure. This paper
shows that and its dual \Dim^\beta(S), the {\it strong
dimension} of with respect to , can be used in conjunction with
randomness to measure the similarity of two probability measures and
on . Specifically, we prove that the {\it divergence formula}
\dim^\beta(R) = \Dim^\beta(R) =\CH(\alpha) / (\CH(\alpha) + \D(\alpha ||
\beta)) holds whenever and are computable, positive
probability measures on and is random with
respect to . In this formula, \CH(\alpha) is the Shannon entropy of
, and \D(\alpha||\beta) is the Kullback-Leibler divergence between
and
Criticality and entanglement in random quantum systems
We review studies of entanglement entropy in systems with quenched
randomness, concentrating on universal behavior at strongly random quantum
critical points. The disorder-averaged entanglement entropy provides insight
into the quantum criticality of these systems and an understanding of their
relationship to non-random ("pure") quantum criticality. The entanglement near
many such critical points in one dimension shows a logarithmic divergence in
subsystem size, similar to that in the pure case but with a different universal
coefficient. Such universal coefficients are examples of universal critical
amplitudes in a random system. Possible measurements are reviewed along with
the one-particle entanglement scaling at certain Anderson localization
transitions. We also comment briefly on higher dimensions and challenges for
the future.Comment: Review article for the special issue "Entanglement entropy in
extended systems" in J. Phys.
Random paths and current fluctuations in nonequilibrium statistical mechanics
An overview is given of recent advances in nonequilibrium statistical
mechanics about the statistics of random paths and current fluctuations.
Although statistics is carried out in space for equilibrium statistical
mechanics, statistics is considered in time or spacetime for nonequilibrium
systems. In this approach, relationships have been established between
nonequilibrium properties such as the transport coefficients, the thermodynamic
entropy production, or the affinities, and quantities characterizing the
microscopic Hamiltonian dynamics and the chaos or fluctuations it may generate.
This overview presents results for classical systems in the escape-rate
formalism, stochastic processes, and open quantum systems
Entanglement, quantum randomness, and complexity beyond scrambling
Scrambling is a process by which the state of a quantum system is effectively
randomized due to the global entanglement that "hides" initially localized
quantum information. In this work, we lay the mathematical foundations of
studying randomness complexities beyond scrambling by entanglement properties.
We do so by analyzing the generalized (in particular R\'enyi) entanglement
entropies of designs, i.e. ensembles of unitary channels or pure states that
mimic the uniformly random distribution (given by the Haar measure) up to
certain moments. A main collective conclusion is that the R\'enyi entanglement
entropies averaged over designs of the same order are almost maximal. This
links the orders of entropy and design, and therefore suggests R\'enyi
entanglement entropies as diagnostics of the randomness complexity of
corresponding designs. Such complexities form a hierarchy between information
scrambling and Haar randomness. As a strong separation result, we prove the
existence of (state) 2-designs such that the R\'enyi entanglement entropies of
higher orders can be bounded away from the maximum. However, we also show that
the min entanglement entropy is maximized by designs of order only logarithmic
in the dimension of the system. In other words, logarithmic-designs already
achieve the complexity of Haar in terms of entanglement, which we also call
max-scrambling. This result leads to a generalization of the fast scrambling
conjecture, that max-scrambling can be achieved by physical dynamics in time
roughly linear in the number of degrees of freedom.Comment: 72 pages, 4 figures. Rewritten version with new title. v3: published
versio
On the concept of complexity in random dynamical systems
We introduce a measure of complexity in terms of the average number of bits
per time unit necessary to specify the sequence generated by the system. In
random dynamical system, this indicator coincides with the rate K of divergence
of nearby trajectories evolving under two different noise realizations.
The meaning of K is discussed in the context of the information theory, and
it is shown that it can be determined from real experimental data. In presence
of strong dynamical intermittency, the value of K is very different from the
standard Lyapunov exponent computed considering two nearby trajectories
evolving under the same randomness. However, the former is much more relevant
than the latter from a physical point of view as illustrated by some numerical
computations for noisy maps and sandpile models.Comment: 35 pages, LaTe
Group Field theory and Tensor Networks: towards a Ryu-Takayanagi formula in full quantum gravity
We establish a dictionary between group field theory (thus, spin networks and
random tensors) states and generalized random tensor networks. Then, we use
this dictionary to compute the R\'{e}nyi entropy of such states and recover the
Ryu-Takayanagi formula, in two different cases corresponding to two different
truncations/approximations, suggested by the established correspondence.Comment: 54 pages, 10 figures; v2: replace figure 1 with a new version.
Matches submitted version. v3: remove Renyi entropy computation on the random
tensor network, focusing on GFT computation and interpretatio
Particle-hole symmetric localization in two dimensions
We revisit two-dimensional particle-hole symmetric sublattice localization
problem, focusing on the origin of the observed singularities in the density of
states at the band center E=0. The most general such system [R. Gade,
Nucl. Phys. B {\bf 398}, 499 (1993)] exhibits critical behavior and has
that diverges stronger than any integrable power-law, while the
special {\it random vector potential model} of Ludwiget al [Phys. Rev. B {\bf
50}, 7526 (1994)] has instead a power-law density of states with a continuously
varying dynamical exponent. We show that the latter model undergoes a dynamical
transition with increasing disorder--this transition is a counterpart of the
static transition known to occur in this system; in the strong-disorder regime,
we identify the low-energy states of this model with the local extrema of the
defining two-dimensional Gaussian random surface. Furthermore, combining this
``surface fluctuation'' mechanism with a renormalization group treatment of a
related vortex glass problem leads us to argue that the asymptotic low
behavior of the density of states in the {\it general} case is , different from earlier prediction of Gade. We also
study the localized phases of such particle-hole symmetric systems and identify
a Griffiths ``string'' mechanism that generates singular power-law
contributions to the low-energy density of states in this case.Comment: 18 pages (two-column PRB format), 10 eps figures include
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