We consider the so-called frog model with random initial configurations. The dynamics of this model are described as follows. Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and these independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start moving in a similar fashion. The aim of this paper is to derive large deviation and concentration bounds for the first passage time at which an active particle reaches a target site.

Denote the Palm measure of a homogeneous Poisson process Hλ with two points 0 and x by 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||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(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.