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 $\lambda_c^{(d)}$ 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 $\lambda^{(3)}_c \geq 0.4494...$ 
Zhenning Kong and Edmund M. Yeh