29 research outputs found
Spread-out percolation in R^d
Let be either or the points of a Poisson process in of
intensity 1. Given parameters and , join each pair of points of
within distance independently with probability . This is the simplest
case of a `spread-out' percolation model studied by Penrose, who showed that,
as , the average degree of the corresponding random graph at the
percolation threshold tends to 1, i.e., the percolation threshold and the
threshold for criticality of the naturally associated branching process
approach one another. Here we show that this result follows immediately from of
a general result of the authors on inhomogeneous random graphs.Comment: 9 pages. Title changed. Minor changes to text, including updated
references to [3]. To appear in Random Structures and Algorithm
Synchronization of moving integrate and fire oscillators
We present a model of integrate and fire oscillators that move on a plane.
The phase of the oscillators evolves linearly in time and when it reaches a
threshold value they fire choosing their neighbors according to a certain
interaction range. Depending on the velocity of the ballistic motion and the
average number of neighbors each oscillator fires to, we identify different
regimes shown in a phase diagram. We characterize these regimes by means of
novel parameters as the accumulated number of contacted neighbors.Comment: 9 pages, 5 figure
A short proof of the Harris-Kesten Theorem
We give a short proof of the fundamental result that the critical probability
for bond percolation in the planar square lattice is equal to 1/2. The lower
bound was proved by Harris, who showed in 1960 that percolation does not occur
at . The other, more difficult, bound was proved by Kesten, who showed
in 1980 that percolation does occur for any .Comment: 17 pages, 9 figures; typos corrected. To appear in the Bulletin of
the London Mathematical Societ
Percolation for D2D networks on street systems
We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold
Disagreement percolation for the hard-sphere model
Disagreement percolation connects a Gibbs lattice gas and i.i.d. site
percolation on the same lattice such that non-percolation implies uniqueness of
the Gibbs measure. This work generalises disagreement percolation to the
hard-sphere model and the Boolean model. Non-percolation of the Boolean model
implies the uniqueness of the Gibbs measure and exponential decay of pair
correlations and finite volume errors. Hence, lower bounds on the critical
intensity for percolation of the Boolean model imply lower bounds on the
critical activity for a (potential) phase transition. These lower bounds
improve upon known bounds obtained by cluster expansion techniques. The proof
uses a novel dependent thinning from a Poisson point process to the hard-sphere
model, with the thinning probability related to a derivative of the free
energy
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