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2021 On the Manhattan pinball problem
Linjun Li
Author Affiliations +
Electron. Commun. Probab. 26: 1-11 (2021). DOI: 10.1214/21-ECP394


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 is true for p>12ε with some ε>0.

Funding Statement

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.


Download Citation

Linjun Li. "On the Manhattan pinball problem." Electron. Commun. Probab. 26 1 - 11, 2021.


Received: 5 October 2020; Accepted: 20 April 2021; Published: 2021
First available in Project Euclid: 5 May 2021

Digital Object Identifier: 10.1214/21-ECP394

Primary: 37A50 , 60K35

Keywords: Localization , Lorentz lattice gas , Manhattan pinball


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