3,027 research outputs found
Langevin process reflected on a partially elastic boundary I
Consider a Langevin process, that is an integrated Brownian motion,
constrained to stay on the nonnegative half-line by a partially elastic
boundary at 0. If the elasticity coefficient of the boundary is greater than or
equal to a critical value (0.16), bounces will not accumulate in a finite time
when the process starts from the origin with strictly positive velocity. We
will show that there exists then a unique entrance law from the boundary with
zero velocity, despite the immediate accumulation of bounces. This result of
uniqueness is in sharp contrast with the literature on deterministic second
order reflection. Our approach uses certain properties of real-valued random
walks and a notion of spatial stationarity which may be of independent
interest.Comment: 30 pages, 1 figure. In this new version, the introduction and the
preliminaries in particular have been rewritten (for a dramatic change
Spatial preferential attachment networks: Power laws and clustering coefficients
We define a class of growing networks in which new nodes are given a spatial
position and are connected to existing nodes with a probability mechanism
favoring short distances and high degrees. The competition of preferential
attachment and spatial clustering gives this model a range of interesting
properties. Empirical degree distributions converge to a limit law, which can
be a power law with any exponent . The average clustering coefficient
of the networks converges to a positive limit. Finally, a phase transition
occurs in the global clustering coefficients and empirical distribution of edge
lengths when the power-law exponent crosses the critical value . Our
main tool in the proof of these results is a general weak law of large numbers
in the spirit of Penrose and Yukich.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1006 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Robustness of scale-free spatial networks
A growing family of random graphs is called robust if it retains a giant
component after percolation with arbitrary positive retention probability. We
study robustness for graphs, in which new vertices are given a spatial position
on the -dimensional torus and are connected to existing vertices with a
probability favouring short spatial distances and high degrees. In this model
of a scale-free network with clustering we can independently tune the power law
exponent of the degree distribution and the rate at which the
connection probability decreases with the distance of two vertices. We show
that the network is robust if , but fails to be robust if
. In the case of one-dimensional space we also show that the network is
not robust if . This implies that robustness of a
scale-free network depends not only on its power-law exponent but also on its
clustering features. Other than the classical models of scale-free networks our
model is not locally tree-like, and hence we need to develop novel methods for
its study, including, for example, a surprising application of the
BK-inequality.Comment: 34 pages, 4 figure
The spread of infections on evolving scale-free networks
We study the contact process on a class of evolving scale-free networks,
where each node updates its connections at independent random times. We give a
rigorous mathematical proof that there is a transition between a phase where
for all infection rates the infection survives for a long time, at least
exponential in the network size, and a phase where for sufficiently small
infection rates extinction occurs quickly, at most like the square root of the
network size. The phase transition occurs when the power-law exponent crosses
the value four. This behaviour is in contrast to that of the contact process on
the corresponding static model, where there is no phase transition, as well as
that of a classical mean-field approximation, which has a phase transition at
power-law exponent three. The new observation behind our result is that
temporal variability of networks can simultaneously increase the rate at which
the infection spreads in the network, and decrease the time which the infection
spends in metastable states.Comment: 17 pages, 1 figur
Excursions of the integral of the Brownian motion
The integrated Brownian motion is sometimes known as the Langevin process.
Lachal studied several excursion laws induced by the latter. Here we follow a
different point of view developed by Pitman for general stationary processes.
We first construct a stationary Langevin process and then determine explicitly
its stationary excursion measure. This is then used to provide new descriptions
of Ito's excursion measure of the Langevin process reflected at a completely
inelastic boundary, which has been introduced recently by Bertoin.Comment: In this second version, some consequent changes of notations and
presentation. The space we work on for Proposition 2 and Theorem 2 changed a
bit (the proofs are unchanged
The scaling limit of uniform random plane maps, via the Ambj{\o}rn-Budd bijection
We prove that a uniform rooted plane map with n edges converges in
distribution after a suitable normalization to the Brownian map for the
Gromov-Hausdorff topology. A recent bijection due to Ambj{\o}rn and Budd allows
to derive this result by a direct coupling with a uniform random
quadrangulation with n faces
A note on Fourier coefficients of Poincar\'e series
We give a short and "soft" proof of the asymptotic orthogonality of Fourier
coefficients of Poincar\'e series for classical modular forms as well as for
Siegel cusp forms, in a qualitative form.Comment: 10 page
Robustness of scale-free spatial networks
A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the d-dimensional torus and are connected to existing vertices with a probability favouring short spatial distances and high degrees. In this model of a scale-free network with clustering we can independently tune the power law exponent tau of the degree distribution and the rate delta d at which the connection probability decreases with the distance of two vertices. We show that the network is robust if tau<2+1/delta, but fails to be robust if tau>3. In the case of one-dimensional space we also show that the network is not robust if tau<2+1/(delta-1). This implies that robustness of a scale-free network depends not only on its power-law exponent but also on its clustering features. Other than the classical models of scale-free networks our model is not locally tree-like, and hence we need to develop novel methods for its study, including, for example, a surprising application of the BK-inequality
Developmental Psychology: A Precursor of Moral Judgment in Human Infants?
Human infants evaluate social interactions well before they can speak, and show a preference for characters that help others over characters that are not cooperative or are hindering
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