3,282 research outputs found
Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian
We prove for the square Fibonacci Hamiltonian that the density of states
measure is absolutely continuous for almost all pairs of small coupling
constants. This is obtained from a new result we establish about the absolute
continuity of convolutions of measures arising in hyperbolic dynamics with
exact-dimensional measures.Comment: 28 pages, to appear in Duke Math.
Poisson-Pinsker factor and infinite measure preserving group actions
We solve the question of the existence of a Poisson-Pinsker factor for
conservative ergodic infinite measure preserving action of a countable amenable
group by proving the following dichotomy: either it has totally positive
Poisson entropy (and is of zero type), or it possesses a Poisson-Pinsker
factor. If G is abelian and the entropy positive, the spectrum is absolutely
continuous (Lebesgue countable if G=\mathbb{Z}) on the whole L^{2}-space in the
first case and in the orthocomplement of the L^{2}-space of the Poisson-Pinsker
factor in the second.Comment: 9 page
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
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Spectral transitions for the square Fibonacci Hamiltonian
We study the spectrum and the density of states measure of the square Fibonacci Hamiltonian. We describe where the transitions from positive-measure to zero-measure spectrum and from absolutely continuous to singular density of states measure occur. This shows in particular that for almost every parameter from some open set, a positive-measure spectrum and a singular density of states measure coexist. This provides the first physically relevant example exhibiting this phenomenon
Asymptotic entropy and green speed for random walks on countable groups
We study asymptotic properties of the Green metric associated with transient
random walks on countable groups. We prove that the rate of escape of the
random walk computed in the Green metric equals its asymptotic entropy. The
proof relies on integral representations of both quantities with the extended
Martin kernel. In the case of finitely generated groups, where this result is
known (Benjamini and Peres [Probab. Theory Related Fields 98 (1994) 91--112]),
we give an alternative proof relying on a version of the so-called fundamental
inequality (relating the rate of escape, the entropy and the logarithmic volume
growth) extended to random walks with unbounded support.Comment: Published in at http://dx.doi.org/10.1214/07-AOP356 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Poisson suspensions and infinite ergodic theory
We investigate ergodic theory of Poisson suspensions. In the process, we
establish close connections between finite and infinite measure preserving
ergodic theory. Poisson suspensions thus provide a new approach to infinite
measure preserving ergodic theory. Fields investigated here are mixing
properties, spectral theory, joinings. We also compare Poisson suspensions to
the apparently similar looking Gaussian dynamical systems.Comment: 18 page
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