Visible lattice points in random walks. (January 2019)

• Record Type: Journal Article
• Authors: Cilleruelo, Javier; Fernández, José L.; Fernández, Pablo
• Is Part Of: European journal of combinatorics. Volume 75(2019)
• Page Start: 92
• Page End: 112
• Publication Date: 2019-01
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2018.08.004
• Languages: English
• ISSNs: 0195-6698

Abstract

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 .