On the Manhattan pinball problem

Abstract

We consider the periodic Manhattan lattice with alternating orientations going north-south 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 for $p>\frac{1}{2}-\epsilon$ with some $\epsilon >0$.

11 pages, 10 figures; added more references in the introduction, simplified the construction for enhancement, added more details in the proof of the main theorem, added appendix A to explain how to adapt the enhancement argument to our sepecific case, fixed typos, main result unchanged