Secrecy coverage in two dimensions

Abstract

Working in the infinite plane R², consider a Poisson process of black points with intensity 1, and an independent Poisson process of red points with intensity λ. We grow a disc around each black point until it hits the nearest red point, resulting in a random configuration A_λ, which is the union of discs centered at the black points. Next, consider a fixed disc of area n in the plane. What is the probability p_λ(n) that this disc is covered by A_λ? We prove that if λ³nlogn = y then, for sufficiently large n, e^-8π2y ≤ p_λ(n) ≤ e^-2π2y/3. The proofs reveal a new and surprising phenomenon, namely, that the obstructions to coverage occur on a wide range of scales.