4,766 research outputs found
Moment asymptotics for multitype branching random walks in random environment
We study a discrete time multitype branching random walk on a finite space
with finite set of types. Particles follow a Markov chain on the spatial space
whereas offspring distributions are given by a random field that is fixed
throughout the evolution of the particles. Our main interest lies in the
averaged (annealed) expectation of the population size, and its long-time
asymptotics. We first derive, for fixed time, a formula for the expected
population size with fixed offspring distributions, which is reminiscent of a
Feynman-Kac formula. We choose Weibull-type distributions with parameter
for the upper tail of the mean number of type particles
produced by an type particle. We derive the first two terms of the
long-time asymptotics, which are written as two coupled variational formulas,
and interpret them in terms of the typical behavior of the system
Asymptotics for the small fragments of the fragmentation at nodes
We consider the fragmentation at nodes of the L\'{e}vy continuous random tree
introduced in a previous paper. In this framework we compute the asymptotic for
the number of small fragments at time . This limit is increasing in
and discontinuous. In the -stable case the fragmentation is
self-similar with index , with and the results are
close to those Bertoin obtained for general self-similar fragmentations but
with an additional assumtion which is not fulfilled here
Power Law Distributions of Offspring and Generation Numbers in Branching Models of Earthquake Triggering
We consider a general stochastic branching process, which is relevant to
earthquakes as well as to many other systems, and we study the distributions of
the total number of offsprings (direct and indirect aftershocks in seismicity)
and of the total number of generations before extinction. We apply our results
to a branching model of triggered seismicity, the ETAS (epidemic-type
aftershock sequence) model. The ETAS model assumes that each earthquake can
trigger other earthquakes (``aftershocks''). An aftershock sequence results in
this model from the cascade of aftershocks of each past earthquake. Due to the
large fluctuations of the number of aftershocks triggered directly by any
earthquake (``fertility''), there is a large variability of the total number of
aftershocks from one sequence to another, for the same mainshock magnitude. We
study the regime where the distribution of fertilities mu is characterized by a
power law ~1/\mu^(1+gamma). For earthquakes, we expect such a power-law
distribution of fertilities with gamma = b/alpha based on the Gutenberg-Richter
magnitude distribution ~10^(-bm) and on the increase ~10^(alpha m) of the
number of aftershocks with the mainshock magnitude m. We derive the asymptotic
distributions p_r(r) and p_g(g) of the total number r of offsprings and of the
total number g of generations until extinction following a mainshock. In the
regime \gamma<2 relevant for earhquakes, for which the distribution of
fertilities has an infinite variance, we find p_r(r)~1/r^(1+1/gamma) and
p_g(g)~1/g^(1+1/(gamma -1)). These predictions are checked by numerical
simulations.Comment: revtex, 12 pages, 2 ps figures. In press in Pure and Applied
Geophysics (2004
Finite-Block-Length Analysis in Classical and Quantum Information Theory
Coding technology is used in several information processing tasks. In
particular, when noise during transmission disturbs communications, coding
technology is employed to protect the information. However, there are two types
of coding technology: coding in classical information theory and coding in
quantum information theory. Although the physical media used to transmit
information ultimately obey quantum mechanics, we need to choose the type of
coding depending on the kind of information device, classical or quantum, that
is being used. In both branches of information theory, there are many elegant
theoretical results under the ideal assumption that an infinitely large system
is available. In a realistic situation, we need to account for finite size
effects. The present paper reviews finite size effects in classical and quantum
information theory with respect to various topics, including applied aspects
A Quantitative Study of Pure Parallel Processes
In this paper, we study the interleaving -- or pure merge -- operator that
most often characterizes parallelism in concurrency theory. This operator is a
principal cause of the so-called combinatorial explosion that makes very hard -
at least from the point of view of computational complexity - the analysis of
process behaviours e.g. by model-checking. The originality of our approach is
to study this combinatorial explosion phenomenon on average, relying on
advanced analytic combinatorics techniques. We study various measures that
contribute to a better understanding of the process behaviours represented as
plane rooted trees: the number of runs (corresponding to the width of the
trees), the expected total size of the trees as well as their overall shape.
Two practical outcomes of our quantitative study are also presented: (1) a
linear-time algorithm to compute the probability of a concurrent run prefix,
and (2) an efficient algorithm for uniform random sampling of concurrent runs.
These provide interesting responses to the combinatorial explosion problem
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