Analytical Lower Bounds on the Critical Density in Continuum Percolation

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 λ
_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 λ
_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 λ
_c
⁽²⁾
≥0.7698. For the three-dimensional case, we obtain λ
_c
⁽³⁾
≥0.4494.