Analytical Lower Bounds on the Critical Density in Continuum Percolation

Details Analytical Lower Bounds on the Critical Density in Continuum Percolation Analytical Lower Bounds on the Critical Density in Continuum Percolation

2006-10-25

arxiv.org/abs/math/0610751v3

Mathematics

Probability

Scientific paper

Percolation theory has become a useful tool for the analysis of large-scale wireless networks. We investigate the fundamental problem of characterizing the critical density $\lambda_c^{(d)}$ for $d$-dimensional Poisson random geometric graphs in continuum percolation theory. By using a probabilistic analysis which incorporates the clustering effect in random geometric graphs, we develop a new class of analytical lower bounds for the critical density $\lambda_c^{(d)}$ in $d$-dimensional Poisson random geometric graphs. The lower bounds are the tightest known to date. In particular, for the two-dimensional case, the analytical lower bound is improved to $\lambda^{(2)}_c \geq 0.7698...$. For the three-dimensional case, we obtain $\lambda^{(3)}_c \geq 0.4494...$

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