14,051 research outputs found
A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model
As well known, cumulant expansion is an alternative way to moment expansion
to fully characterize probability distributions provided all the moments exist.
If this is not the case, the so called escort mean values (or q-moments) have
been proposed to characterize probability densities with divergent moments [C.
Tsallis et al, J. Math. Phys 50, 043303 (2009)]. We introduce here a new
mathematical object, namely the q-cumulants, which, in analogy to the
cumulants, provide an alternative characterization to that of the q-moments for
the probability densities. We illustrate this new scheme on a recently proposed
family of scale-invariant discrete probabilistic models [A. Rodriguez et al, J.
Stat. Mech. (2008) P09006; R. Hanel et al, Eur. Phys. J. B 72, 263268 (2009)]
having q-Gaussians as limiting probability distributions
Geometry of River Networks II: Distributions of Component Size and Number
The structure of a river network may be seen as a discrete set of nested
sub-networks built out of individual stream segments. These network components
are assigned an integral stream order via a hierarchical and discrete ordering
method. Exponential relationships, known as Horton's laws, between stream order
and ensemble-averaged quantities pertaining to network components are observed.
We extend these observations to incorporate fluctuations and all higher moments
by developing functional relationships between distributions. The relationships
determined are drawn from a combination of theoretical analysis, analysis of
real river networks including the Mississippi, Amazon and Nile, and numerical
simulations on a model of directed, random networks. Underlying distributions
of stream segment lengths are identified as exponential. Combinations of these
distributions form single-humped distributions with exponential tails, the sums
of which are in turn shown to give power law distributions of stream lengths.
Distributions of basin area and stream segment frequency are also addressed.
The calculations identify a single length-scale as a measure of size
fluctuations in network components. This article is the second in a series of
three addressing the geometry of river networks.Comment: 16 pages, 13 figures, 4 tables, Revtex4, submitted to PR
Poisson approximation of the length spectrum of random surfaces
Multivariate Poisson approximation of the length spectrum of random surfaces
is studied by means of the Chen-Stein method. This approach delivers simple and
explicit error bounds in Poisson limit theorems. They are used to prove that
Poisson approximation applies to curves of length up to order
with being the genus of the surface.Comment: 22 pages, 2 figures. To appear in Indiana Univ. Math.
Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web
We study the topology of the Megaparsec Cosmic Web in terms of the
scale-dependent Betti numbers, which formalize the topological information
content of the cosmic mass distribution. While the Betti numbers do not fully
quantify topology, they extend the information beyond conventional cosmological
studies of topology in terms of genus and Euler characteristic. The richer
information content of Betti numbers goes along the availability of fast
algorithms to compute them.
For continuous density fields, we determine the scale-dependence of Betti
numbers by invoking the cosmologically familiar filtration of sublevel or
superlevel sets defined by density thresholds. For the discrete galaxy
distribution, however, the analysis is based on the alpha shapes of the
particles. These simplicial complexes constitute an ordered sequence of nested
subsets of the Delaunay tessellation, a filtration defined by the scale
parameter, . As they are homotopy equivalent to the sublevel sets of
the distance field, they are an excellent tool for assessing the topological
structure of a discrete point distribution. In order to develop an intuitive
understanding for the behavior of Betti numbers as a function of , and
their relation to the morphological patterns in the Cosmic Web, we first study
them within the context of simple heuristic Voronoi clustering models.
Subsequently, we address the topology of structures emerging in the standard
LCDM scenario and in cosmological scenarios with alternative dark energy
content. The evolution and scale-dependence of the Betti numbers is shown to
reflect the hierarchical evolution of the Cosmic Web and yields a promising
measure of cosmological parameters. We also discuss the expected Betti numbers
as a function of the density threshold for superlevel sets of a Gaussian random
field.Comment: 42 pages, 14 figure
A glimpse of the conformal structure of random planar maps
We present a way to study the conformal structure of random planar maps. The
main idea is to explore the map along an SLE (Schramm--Loewner evolution)
process of parameter and to combine the locality property of the
SLE_{6} together with the spatial Markov property of the underlying lattice in
order to get a non-trivial geometric information. We follow this path in the
case of the conformal structure of random triangulations with a boundary. Under
a reasonable assumption called (*) that we have unfortunately not been able to
verify, we prove that the limit of uniformized random planar triangulations has
a fractal boundary measure of Hausdorff dimension almost surely.
This agrees with the physics KPZ predictions and represents a first step
towards a rigorous understanding of the links between random planar maps and
the Gaussian free field (GFF).Comment: To appear in Commun. Math. Phy
Intrinsically-generated fluctuating activity in excitatory-inhibitory networks
Recurrent networks of non-linear units display a variety of dynamical regimes
depending on the structure of their synaptic connectivity. A particularly
remarkable phenomenon is the appearance of strongly fluctuating, chaotic
activity in networks of deterministic, but randomly connected rate units. How
this type of intrinsi- cally generated fluctuations appears in more realistic
networks of spiking neurons has been a long standing question. To ease the
comparison between rate and spiking networks, recent works investigated the
dynami- cal regimes of randomly-connected rate networks with segregated
excitatory and inhibitory populations, and firing rates constrained to be
positive. These works derived general dynamical mean field (DMF) equations
describing the fluctuating dynamics, but solved these equations only in the
case of purely inhibitory networks. Using a simplified excitatory-inhibitory
architecture in which DMF equations are more easily tractable, here we show
that the presence of excitation qualitatively modifies the fluctuating activity
compared to purely inhibitory networks. In presence of excitation,
intrinsically generated fluctuations induce a strong increase in mean firing
rates, a phenomenon that is much weaker in purely inhibitory networks.
Excitation moreover induces two different fluctuating regimes: for moderate
overall coupling, recurrent inhibition is sufficient to stabilize fluctuations,
for strong coupling, firing rates are stabilized solely by the upper bound
imposed on activity, even if inhibition is stronger than excitation. These
results extend to more general network architectures, and to rate networks
receiving noisy inputs mimicking spiking activity. Finally, we show that
signatures of the second dynamical regime appear in networks of
integrate-and-fire neurons
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