Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices

For connectivity of \emph{random geometric graphs}, where there is no density for underlying distribution of the vertices, we consider $n$ i.i.d. \emph{Cantor} distributed points on $[0,1]$. We show that for this random geometric graph, the connectivity threshold $R_n$, converges almost surely to a constant $1-2\varphi$ where $0<\varphi <1\mathrm{/}2$, which for the standard Cantor distribution is 1/3. We also show that $| R_n - (1 - 2 \phi) |1 \sim 2 , C(\phi) , n^{-1/d{\phi}}$where$C(\phi) > 0$isaconstantand$d_{\phi} :=

• {\log 2}/{\log \phi}$ is the \emph{Hausdorff dimension} of the generalized Cantor set with parameter $\varphi$.