Asymptotic distribution of critical transmission radius for greedy forward routing

Consider a random multihop wireless network represented by a Poisson point process over a unit-area disk with mean n. Let ø_n denote its critical transmission radius for its greedy forward routing. Recently, asymptotic bounds on ø_n have been progressively improved. However, the precise asymptotic probability distribution of ø_n remains open. In this paper, we settle this open problem. Specifically, let σ = 2π/3 - √3/2. Then for any constant c, the asymptotic probability of equation is proved to be exactly exp (-(1/σ/π-1/3-π/2σ)e^-c).