2,071 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
A sharper threshold for bootstrap percolation in two dimensions
Two-dimensional bootstrap percolation is a cellular automaton in which sites
become 'infected' by contact with two or more already infected nearest
neighbors. We consider these dynamics, which can be interpreted as a monotone
version of the Ising model, on an n x n square, with sites initially infected
independently with probability p. The critical probability p_c is the smallest
p for which the probability that the entire square is eventually infected
exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim
\pi^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the
second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining
it up to a poly(log log n)-factor. The exponent -3/2 corrects numerical
predictions from the physics literature.Comment: 21 page
Scaling and Inverse Scaling in Anisotropic Bootstrap percolation
In bootstrap percolation it is known that the critical percolation threshold
tends to converge slowly to zero with increasing system size, or, inversely,
the critical size diverges fast when the percolation probability goes to zero.
To obtain higher-order terms (that is, sharp and sharper thresholds) for the
percolation threshold in general is a hard question. In the case of
two-dimensional anisotropic models, sometimes correction terms can be obtained
from inversion in a relatively simple manner.Comment: Contribution to the proceedings of the 2013 EURANDOM workshop
Probabilistic Cellular Automata: Theory, Applications and Future
Perspectives, equation typo corrected, constant of generalisation correcte
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