1,704 research outputs found
Percolation in the Secrecy Graph
The secrecy graph is a random geometric graph which is intended to model the
connectivity of wireless networks under secrecy constraints. Directed edges in
the graph are present whenever a node can talk to another node securely in the
presence of eavesdroppers, which, in the model, is determined solely by the
locations of the nodes and eavesdroppers. In the case of infinite networks, a
critical parameter is the maximum density of eavesdroppers that can be
accommodated while still guaranteeing an infinite component in the network,
i.e., the percolation threshold. We focus on the case where the locations of
the nodes and eavesdroppers are given by Poisson point processes, and present
bounds for different types of percolation, including in-, out- and undirected
percolation.Comment: 22 pages, 3 figure
Two-Dimensional Critical Percolation: The Full Scaling Limit
We use SLE(6) paths to construct a process of continuum nonsimple loops in
the plane and prove that this process coincides with the full continuum scaling
limit of 2D critical site percolation on the triangular lattice -- that is, the
scaling limit of the set of all interfaces between different clusters. Some
properties of the loop process, including conformal invariance, are also
proved.Comment: 45 pages, 12 figures. This is a revised version of math.PR/0504036
without the appendice
Theta-point polymers in the plane and Schramm-Loewner evolution
We study the connection between polymers at the theta temperature on the
lattice and Schramm-Loewner chains with constant step length in the continuum.
The latter realize a useful algorithm for the exact sampling of tricritical
polymers, where finite-chain effects are excluded. The driving function
computed from the lattice model via a radial implementation of the zipper
method is shown to converge to Brownian motion of diffusivity kappa=6 for large
times. The distribution function of an internal portion of walk is well
approximated by that obtained from Schramm-Loewner chains. The exponent of the
correlation length nu and the leading correction-to scaling exponent Delta_1
measured in the continuum are compatible with nu=4/7 (predicted for the theta
point) and Delta_1=72/91 (predicted for percolation). Finally, we compute the
shape factor and the asphericity of the chains, finding surprising accord with
the theta-point end-to-end values.Comment: 8 pages, 6 figure
Continuum Nonsimple Loops and 2D Critical Percolation
Substantial progress has been made in recent years on the 2D critical
percolation scaling limit and its conformal invariance properties. In
particular, chordal SLE6 (the Stochastic Loewner Evolution with parameter k=6)
was, in the work of Schramm and of Smirnov, identified as the scaling limit of
the critical percolation ``exploration process.'' In this paper we use that and
other results to construct what we argue is the full scaling limit of the
collection of all closed contours surrounding the critical percolation clusters
on the 2D triangular lattice. This random process or gas of continuum nonsimple
loops in the plane is constructed inductively by repeated use of chordal SLE6.
These loops do not cross but do touch each other -- indeed, any two loops are
connected by a finite ``path'' of touching loops.Comment: 16 pages, 3 figure
Line-of-sight percolation
Given , let be the graph with vertex set
in which two vertices are joined if they agree in one coordinate and differ by
at most in the other. (Thus is precisely .) Let
be the critical probability for site percolation in
. Extending recent results of Frieze, Kleinberg, Ravi and
Debany, we show that \lim_{\omega\to\infty} \omega\pc(\omega)=\log(3/2). We
also prove analogues of this result on the -by- grid and in higher
dimensions, the latter involving interesting connections to Gilbert's continuum
percolation model. To prove our results, we explore the component of the origin
in a certain non-standard way, and show that this exploration is well
approximated by a certain branching random walk.Comment: Revised and expanded (section 2.3 added). To appear in Combinatorics,
Probability and Computing. 27 pages, 4 figure
Existence of an unbounded vacant set for subcritical continuum percolation
We consider the Poisson Boolean percolation model in , where
the radii of each ball is independently chosen according to some probability
measure with finite second moment. For this model, we show that the two
thresholds, for the existence of an unbounded occupied and an unbounded vacant
component, coincide. This complements a recent study of the sharpness of the
phase transition in Poisson Boolean percolation by the same authors. As a
corollary it follows that for Poisson Boolean percolation in ,
for any , finite moment of order is both necessary and sufficient
for the existence of a nontrivial phase transition for the vacant set.Comment: 9 page
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