Disjoint random needles (Q1612307)

Consider a Poisson point process on the plane of intensity \(\rho\). Every point of the process is the center of an isotropic segment (needle) of length \(2\ell\), such that orientations of different segments are independent of each other and of their positions. Let \(D(R)\) denote the cardinality of the largest disjoint subset of needles such that their middle points fall in the square \([-R,R]^2\). A subadditivity argument implies the existence of the limit \(\varphi(\rho,\ell)=\lim_{R\to\infty} R^{-2}D(R)\). The main result of the paper establishes existence of a number \(K\) such that \(K^{-1}\min(\rho,\sqrt{\rho}/\ell)\leq\varphi(\rho,\ell)\leq K\min(\rho,\sqrt{\rho}/\ell)\).