On the Manhattan pinball problem

Abstract

We consider the periodic Manhattan lattice with alternating orientations going northsouth and east-west. Place obstructions on vertices independently with probability 0 < p < 1. A particle is moving on the edges with unit speed following the orientation of the lattice and it will turn only when encountering an obstruction. The problem is that for which value of p is the trajectory of the particle closed almost surely. We prove this is true for p > 1 2 − ε with some ε > 0.

Selected problems in probability theory

“…It follows as in Section 3.2 that θ(q) = 0 for q ≥ 1 2 , and it has been proved by Li in [26] that there exists > 0 such that θ(q) = 0 when q > 1 2 − . The proof uses the method of enhancements; see [1], [11,Sect.…”