Visible lattice points in random walks. (January 2019) - Cilleruelo, Javier
Abstract: We consider the possible visits to visible points of a random walker moving up and right in the integer lattice (with probability α and 1 − α, respectively), and starting from the origin. We show that, almost surely, the asymptotic proportion of strings of k consecutive visible lattice points visited by such an α -random walk is a certain constant c k ( α ), which is actually an (explicitly computable) polynomial in α of degree 2 ⌊ ( k − 1 ) ∕ 2 ⌋ . For k = 1, this gives that, almost surely, the asymptotic proportion of time the random walker is visible from the origin is c 1 ( α ) = 6 ∕ π 2, independently of α .