Large Deviations for the Graph Distance in Supercritical Continuum Percolation

Abstract

Denote the Palm measure of a homogeneous Poisson processHλwith two points 0 andxby P0,x. We prove that there exists a constant μ ≥ 1 such that P0,x(D(0,x) / μ||x||2∉ (1 − ε, 1 + ε) | 0,x∈C∞) exponentially decreases when ||x||2tends to ∞, whereD(0,x) is the graph distance between 0 andxin the infinite componentC∞of the random geometric graphG(Hλ; 1). We derive a large deviation inequality for an asymptotic shape result. Our results have applications in many fields and especially in wireless sensor networks.