Visibility properties of lattice points in multiple random walks (Q6547995)

Consider \(n\) independent random walks on \({\mathbb{N}}^k\), with arbitrary probability \(\alpha_i\) of going in the \(i\)th coordinate direction. A lattice point is \textit{visible} if there is no other lattice point between it and the origin; that is, if the GCD of its coordinates is 1. The author computes the limiting density of the set of times at which all \(k\) walks, or exactly \(m\) walks, are simultaneously visible. The density approaches this limit almost surely. The proofs use results from number theory as well as probability.