110 research outputs found
Thermal corpuscular black holes
We study the corpuscular model of an evaporating black hole consisting of a
specific quantum state for a large number of self-confined bosons. The
single-particle spectrum contains a discrete ground state of energy
(corresponding to toy gravitons forming the black hole), and a gapless
continuous spectrum (to accommodate for the Hawking radiation with energy
). Each constituent is in a superposition of the ground state and a
Planckian distribution at the expected Hawking temperature in the continuum. We
first find that, assuming the Hawking radiation is the leading effect of the
internal scatterings, the corresponding -particle state can be collectively
described by a single-particle wave-function given by a superposition of a
total ground state with energy and a Planckian distribution for
at the same Hawking temperature. From this collective state, we compute the
partition function and obtain an entropy which reproduces the usual area law
with a logarithmic correction precisely related with the Hawking component. By
means of the horizon wave-function for the system, we finally show the
backreaction of modes with reduces the Hawking flux. Both
corrections, to the entropy and to the Hawking flux, suggest the evaporation
properly stops for vanishing mass, if the black hole is in this particular
quantum state.Comment: PDFLaTeX, 15 pages, 2 figure. Version to appear in PR
Consistent Cosmic Microwave Background Spectra from Quantum Depletion
Following a new quantum cosmological model proposed by Dvali and Gomez, we
quantitatively investigate possible modifications to the Hubble parameter and
following corrections to the cosmic microwave background spectrum. In this
model, scalar and tensor perturbations are generated by the quantum depletion
of the background inflaton and graviton condensate respectively. We show how
the inflaton mass affects the power spectra and the tensor-to-scalar ratio.
Masses approaching the Planck scale would lead to strong deviations, while
standard spectra are recovered for an inflaton mass much smaller than the
Planck mass.Comment: 23 pages, 9 figures; revised version to match published versio
Black holes as self-sustained quantum states, and Hawking radiation
We employ the recently proposed formalism of the "horizon wave-function" to
investigate the emergence of a horizon in models of black holes as
Bose-Einstein condensates of gravitons. We start from the Klein-Gordon equation
for a massless scalar (toy graviton) field coupled to a static matter current.
The (spherically symmetric) classical field reproduces the Newtonian potential
generated by the matter source, and the corresponding quantum state is given by
a coherent superposition of scalar modes with continuous occupation number.
Assuming an attractive self-interaction that allows for bound states, one finds
that (approximately) only one mode is allowed, and the system can be confined
in a region of the size of the Schwarzschild radius. This radius is then shown
to correspond to a proper horizon, by means of the horizon wave-function of the
quantum system, with an uncertainty in size naturally related to the expected
typical energy of Hawking modes. In particular, this uncertainty decreases for
larger black hole mass (with larger number of light scalar quanta), in
agreement with semiclassical expectations, a result which does not hold for a
single very massive particle. We finally speculate that a phase transition
should occur during the gravitational collapse of a star, ideally represented
by a static matter current and Newtonian potential, that leads to a black hole,
again ideally represented by the condensate of toy gravitons, and suggest an
effective order parameter that could be used to investigate this transition.Comment: 25 pages, 6 figures. Improved text and typos fixed. Final version to
appear in PR
Thermal BEC black holes
We review some features of BEC models of black holes obtained by means of the
HWF formalism. We consider the KG equation for a toy graviton field coupled to
a static matter current in spherical symmetry. The classical field reproduces
the Newtonian potential generated by the matter source, while the corresponding
quantum state is given by a coherent superposition of scalar modes with
continuous occupation number. An attractive self-interaction is needed for
bound states to form, so that (approximately) one mode is allowed, and the
system of N bosons can be self-confined in a volume of the size of the
Schwarzschild radius. The HWF is then used to show that the radius of such a
system corresponds to a proper horizon. The uncertainty in the size of the
horizon is related to the typical energy of Hawking modes: it decreases with
the increasing of the black hole mass (larger number of gravitons), in
agreement with semiclassical calculations and different from a single very
massive particle. The spectrum contains a discrete ground state of energy
(the bosons forming the black hole), and a continuous spectrum with energy
(representing the Hawking radiation and modelled with a Planckian
distribution at the expected Hawking temperature). The -particle state can
be collectively described by a single-particle wave-function given by a
superposition of a total ground state with energy and a Planckian
distribution for at the same Hawking temperature. The partition
function is then found to yield the usual area law for the entropy, with a
logarithmic correction related with the Hawking component. The backreaction of
modes with is also shown to reduce the Hawking flux and the
evaporation properly stops for vanishing mass.Comment: 30 pages, pdflatex with 6 figures. Review paper prepared for Entropy
special issue "Entropy in Quantum Gravity and Quantum Cosmology
Space-Efficient Substring Occurrence Estimation
In this paper we study the problem of estimating the number of occurrences of substrings in textual data: A text T on some alphabet Σ=[σ] of length n is preprocessed and an index I is built. The index is used in lieu of the text to answer queries of the form Count≈(P), returning an approximated number of the occurrences of an arbitrary pattern P as a substring of T. The problem has its main application in selectivity estimation related to the LIKE predicate in textual databases. Our focus is on obtaining an algorithmic solution with guaranteed error rates and small footprint. To achieve that, we first enrich previous work in the area of compressed text-indexing providing an optimal data structure that, for a given additive error ℓ≥1, requires Θnℓlogσ bits. We also approach the issue of guaranteeing exact answers for sufficiently frequent patterns, providing a data structure whose size scales with the amount of such patterns. Our theoretical findings are supported by experiments showing the practical impact of our data structures
The final stage of gravitationally collapsed thick matter layers
In the presence of a minimal length physical objects cannot collapse to an
infinite density, singular, matter point. In this note we consider the possible
final stage of the gravitational collapse of "thick" matter layers. The energy
momentum tensor we choose to model these shell-like objects is a proper
modification of the source for "non-commutative geometry inspired", regular
black holes. By using higher momenta of Gaussian distribution to localize
matter at finite distance from the origin, we obtain new solutions of the
Einstein's equation which smoothly interpolates between Minkowski geometry near
the center of the shell and Schwarzschild spacetime far away from the matter
layer. The metric is curvature singularity free. Black hole type solutions
exist only for "heavy" shells, i.e. , where is the mass of
the extremal configuration. We determine the Hawking temperature and a modified
Area Law taking into account the extended nature of the source.Comment: v2: 13 pages, 5 figures (1 figure added), text edited, additional
references, in press on the special issue "Experimental Tests of Quantum
Gravity and Exotic Quantum Field Theory Effects'' of Adv. High En. Phy
Advanced rank/select data structures: succinctness, bounds and applications.
The thesis explores new theoretical results and applications of rank and select data structures. Given a string, select(c, i) gives the position of the ith occurrence of character c in the string, while rank(c, p) counts the number of instances of character c on the left of position p.
Succinct rank/select data structures are space-efficient versions of standard ones, designed to keep data compressed and at the same time answer to queries rapidly. They are at the basis of more involved compressed and succinct data structures which in turn are motivated by the nowadays need to analyze and operate on massive data sets quickly, where space efficiency is crucial. The thesis builds up on the state of the art left by years of study and produces results on multiple fronts.
Analyzing binary succinct data structures and their link with predecessor data structures, we integrate data structures for the latter problem in the former. The result is a data structure which outperforms the one of Patrascu 08 in a range of cases which were not studied before, namely when the lower bound for predecessor do not apply and constant-time rank is not feasible.
Further, we propose the first lower bound for succinct data structures on generic strings, achieving a linear trade-off between time for rank/select execution and additional space (w.r.t. to the plain data) needed by the data structure. The proposal addresses systematic data structures, namely those that only access the underlying string through ADT calls and do not encode it directly.
Also, we propose a matching upper bound that proves the tightness of our lower bound.
Finally, we apply rank/select data structures to the substring counting problem, where we seek to preprocess a text and generate a summary data structure which is stored in lieu of the text and answers to substring counting queries with additive error. The results include a theory-proven optimal data structure with generic additive error and a data structure that errs only on infrequent patterns with significative practical space gains
More Haste, Less Waste: Lowering the Redundancy in Fully Indexable Dictionaries
We consider the problem of representing, in a compressed format, a bit-vector
of bits with 1s, supporting the following operations, where : returns the number of occurrences of bit in the
prefix ; returns the position of the th occurrence
of bit in . Such a data structure is called \emph{fully indexable
dictionary (FID)} [Raman et al.,2007], and is at least as powerful as
predecessor data structures. Our focus is on space-efficient FIDs on the
\textsc{ram} model with word size and constant time for all
operations, so that the time cost is independent of the input size. Given the
bitstring to be encoded, having length and containing ones, the
minimal amount of information that needs to be stored is . The state of the art in building a FID for is
given in [Patrascu,2008] using
bits, to support the operations in time. Here, we propose a parametric
data structure exhibiting a time/space trade-off such that, for any real
constants , it
uses B(n,m) + O(n^{1+\delta} + n (\frac{m}{n^s})^\eps) bits and performs
all the operations in time O(s\delta^{-1} + \eps^{-1}). The improvement is
twofold: our redundancy can be lowered parametrically and, fixing ,
we get a constant-time FID whose space is B(n,m) + O(m^\eps/\poly{n}) bits,
for sufficiently large . This is a significant improvement compared to the
previous bounds for the general case
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