Percolation on the Signal to Interference Ratio Graph with Fading

A wireless communication network is considered where any two nodes are connected if the signal-to-interference ratio (SIR) between them is greater than a threshold. We consider the the path-loss plus fading model of wireless signal propagation. Assuming that the nodes of the wireless network are distributed as a Poisson point process (PPP), percolation (formation of an unbounded connected cluster) on the resulting SIR graph is studied as a function of the density of the PPP. We study the super critical regime of percolation and show that for a small enough threshold, there exists a closed interval of densities for which percolation happens with non-zero probability.