We consider the periodic Manhattan lattice with alternating orientations going north-south and east-west. Place obstructions on vertices independently with probability . 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 with some .
The author is partially supported by NSF grant DMS-1757479.
The author thanks Lingfu Zhang for telling the problem to him and having several discussions. The author also thanks Professor Jian Ding for useful comments on writing and thanks Professor Geoffrey Grimmett for reading the early draft and giving helpful suggestions.
"On the Manhattan pinball problem." Electron. Commun. Probab. 26 1 - 11, 2021. https://doi.org/10.1214/21-ECP394