Large deviations for the graph distance in supercritical continuum percolation

Abstract

Denote the Palm measure of a homogeneous Poisson process H_λ with two points 0 and x by P_0,x. We prove that there exists a constant μ ≥ 1 such that P_0,x(D(0, x) / μ||x||₂ ∉ (1 - ε, 1 + ε) | 0, x ∈ C_∞) exponentially decreases when ||x||₂ tends to ∞, where D(0, x) is the graph distance between 0 and x in the infinite component C_∞ of the random geometric graph G(H_λ; 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.