71,682 research outputs found
Sharp threshold for percolation on expanders
We study the appearance of the giant component in random subgraphs of a given
large finite graph G=(V,E) in which each edge is present independently with
probability p. We show that if G is an expander with vertices of bounded
degree, then for any c in ]0,1[, the property that the random subgraph contains
a giant component of size c|V| has a sharp threshold.Comment: Published in at http://dx.doi.org/10.1214/10-AOP610 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Connectivity in Sub-Poisson Networks
We consider a class of point processes (pp), which we call {\em sub-Poisson};
these are pp that can be directionally-convexly () dominated by some
Poisson pp. The order has already been shown useful in comparing various
point process characteristics, including Ripley's and correlation functions as
well as shot-noise fields generated by pp, indicating in particular that
smaller in the order processes exhibit more regularity (less clustering,
less voids) in the repartition of their points. Using these results, in this
paper we study the impact of the ordering of pp on the properties of two
continuum percolation models, which have been proposed in the literature to
address macroscopic connectivity properties of large wireless networks. As the
first main result of this paper, we extend the classical result on the
existence of phase transition in the percolation of the Gilbert's graph (called
also the Boolean model), generated by a homogeneous Poisson pp, to the class of
homogeneous sub-Poisson pp. We also extend a recent result of the same nature
for the SINR graph, to sub-Poisson pp. Finally, as examples we show that the
so-called perturbed lattices are sub-Poisson. More generally, perturbed
lattices provide some spectrum of models that ranges from periodic grids,
usually considered in cellular network context, to Poisson ad-hoc networks, and
to various more clustered pp including some doubly stochastic Poisson ones.Comment: 8 pages, 10 figures, to appear in Proc. of Allerton 2010. For an
extended version see http://hal.inria.fr/inria-00497707 version
Algebraic aspects of increasing subsequences
We present a number of results relating partial Cauchy-Littlewood sums,
integrals over the compact classical groups, and increasing subsequences of
permutations. These include: integral formulae for the distribution of the
longest increasing subsequence of a random involution with constrained number
of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as
new proofs of old formulae; relations of these expressions to orthogonal
polynomials on the unit circle; and explicit bases for invariant spaces of the
classical groups, together with appropriate generalizations of the
straightening algorithm.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references,
other minor change
Universality of trap models in the ergodic time scale
Consider a sequence of possibly random graphs , ,
whose vertices's have i.i.d. weights with a distribution
belonging to the basin of attraction of an -stable law, .
Let , , be a continuous time simple random walk on which
waits a \emph{mean} exponential time at each vertex . Under
considerably general hypotheses, we prove that in the ergodic time scale this
trap model converges in an appropriate topology to a -process. We apply this
result to a class of graphs which includes the hypercube, the -dimensional
torus, , random -regular graphs and the largest component of
super-critical Erd\"os-R\'enyi random graphs
Clustering comparison of point processes with applications to random geometric models
In this chapter we review some examples, methods, and recent results
involving comparison of clustering properties of point processes. Our approach
is founded on some basic observations allowing us to consider void
probabilities and moment measures as two complementary tools for capturing
clustering phenomena in point processes. As might be expected, smaller values
of these characteristics indicate less clustering. Also, various global and
local functionals of random geometric models driven by point processes admit
more or less explicit bounds involving void probabilities and moment measures,
thus aiding the study of impact of clustering of the underlying point process.
When stronger tools are needed, directional convex ordering of point processes
happens to be an appropriate choice, as well as the notion of (positive or
negative) association, when comparison to the Poisson point process is
considered. We explain the relations between these tools and provide examples
of point processes admitting them. Furthermore, we sketch some recent results
obtained using the aforementioned comparison tools, regarding percolation and
coverage properties of the Boolean model, the SINR model, subgraph counts in
random geometric graphs, and more generally, U-statistics of point processes.
We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips
random complexes generated by stationary point processes. A general observation
is that many of the results derived previously for the Poisson point process
generalise to some "sub-Poisson" processes, defined as those clustering less
than the Poisson process in the sense of void probabilities and moment
measures, negative association or dcx-ordering.Comment: 44 pages, 4 figure
Elementary Excitation Modes in a Granular Glass above Jamming
The dynamics of granular media in the jammed, glassy region is described in
terms of "modes", by applying a Principal Component Analysis (PCA) to the
covariance matrix of the position of individual grains. We first demonstrate
that this description is justified and gives sensible results in a regime of
time/densities such that a metastable state can be observed on long enough
timescale to define the reference configuration. For small enough times/system
sizes, or at high enough packing fractions, the spectral properties of the
covariance matrix reveals large, collective fluctuation modes that cannot be
explained by a Random Matrix benchmark where these correlations are discarded.
We then present a first attempt to find a link between the softest modes of the
covariance matrix during a certain "quiet" time interval and the spatial
structure of the rearrangement event that ends this quiet period. The motion
during these cracks is indeed well explained by the soft modes of the dynamics
before the crack, but the number of cracks preceded by a "quiet" period
strongly reduces when the system unjams, questioning the relevance of a
description in terms of modes close to the jamming transition, at least for
frictional grains.Comment: 11 pages, 10 figure
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