1,207 research outputs found
Exact Site Percolation Thresholds Using the Site-to-Bond and Star-Triangle Transformations
I construct a two-dimensional lattice on which the inhomogeneous site
percolation threshold is exactly calculable and use this result to find two
more lattices on which the site thresholds can be determined. The primary
lattice studied here, the ``martini lattice'', is a hexagonal lattice with
every second site transformed into a triangle. The site threshold of this
lattice is found to be , while the others have and
. This last solution suggests a possible approach to establishing
the bound for the hexagonal site threshold, . To derive these
results, I solve a correlated bond problem on the hexagonal lattice by use of
the star-triangle transformation and then, by a particular choice of
correlations, solve the site problem on the martini lattice.Comment: 12 pages, 10 figures. Submitted to Physical Review
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
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