6,473 research outputs found
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
On directed interacting animals and directed percolation
We study the phase diagram of fully directed lattice animals with
nearest-neighbour interactions on the square lattice. This model comprises
several interesting ensembles (directed site and bond trees, bond animals,
strongly embeddable animals) as special cases and its collapse transition is
equivalent to a directed bond percolation threshold. Precise estimates for the
animal size exponents in the different phases and for the critical fugacities
of these special ensembles are obtained from a phenomenological renormalization
group analysis of the correlation lengths for strips of width up to n=17. The
crossover region in the vicinity of the collapse transition is analyzed in
detail and the crossover exponent is determined directly from the
singular part of the free energy. We show using scaling arguments and an exact
relation due to Dhar that is equal to the Fisher exponent
governing the size distribution of large directed percolation clusters.Comment: 23 pages, 3 figures; J. Phys. A 35 (2002) 272
k-core (bootstrap) percolation on complex networks: Critical phenomena and nonlocal effects
We develop the theory of the k-core (bootstrap) percolation on uncorrelated
random networks with arbitrary degree distributions. We show that the k-core
percolation is an unusual, hybrid phase transition with a jump emergence of the
k-core as at a first order phase transition but also with a critical
singularity as at a continuous transition. We describe the properties of the
k-core, explain the meaning of the order parameter for the k-core percolation,
and reveal the origin of the specific critical phenomena. We demonstrate that a
so-called ``corona'' of the k-core plays a crucial role (corona is a subset of
vertices in the k-core which have exactly k neighbors in the k-core). It turns
out that the k-core percolation threshold is at the same time the percolation
threshold of finite corona clusters. The mean separation of vertices in corona
clusters plays the role of the correlation length and diverges at the critical
point. We show that a random removal of even one vertex from the k-core may
result in the collapse of a vast region of the k-core around the removed
vertex. The mean size of this region diverges at the critical point. We find an
exact mapping of the k-core percolation to a model of cooperative relaxation.
This model undergoes critical relaxation with a divergent rate at some critical
moment.Comment: 11 pages, 8 figure
Regularity scalable image coding based on wavelet singularity detection
In this paper, we propose an adaptive algorithm for scalable wavelet image coding, which is based on the general feature, the regularity, of images. In pattern recognition or computer vision, regularity of images is estimated from the oriented wavelet coefficients and quantified by the Lipschitz exponents. To estimate the Lipschitz exponents, evaluating the interscale evolution of the wavelet transform modulus sum (WTMS) over the directional cone of influence was proven to be a better approach than tracing the wavelet transform modulus maxima (WTMM). This is because the irregular sampling nature of the WTMM complicates the reconstruction process. Moreover, examples were found to show that the WTMM representation cannot uniquely characterize a signal. It implies that the reconstruction of signal from its WTMM may not be consistently stable. Furthermore, the WTMM approach requires much more computational effort. Therefore, we use the WTMS approach to estimate the regularity of images from the separable wavelet transformed coefficients. Since we do not concern about the localization issue, we allow the decimation to occur when we evaluate the interscale evolution. After the regularity is estimated, this information is utilized in our proposed adaptive regularity scalable wavelet image coding algorithm. This algorithm can be simply embedded into any wavelet image coders, so it is compatible with the existing scalable coding techniques, such as the resolution scalable and signal-to-noise ratio (SNR) scalable coding techniques, without changing the bitstream format, but provides more scalable levels with higher peak signal-to-noise ratios (PSNRs) and lower bit rates. In comparison to the other feature-based wavelet scalable coding algorithms, the proposed algorithm outperforms them in terms of visual perception, computational complexity and coding efficienc
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