On the Manhattan pinball problem

TL;DR

This paper investigates the behavior of a particle moving on a periodic Manhattan lattice with randomly placed obstructions, establishing conditions under which its trajectory is almost surely closed.

Contribution

It proves that for obstruction probabilities greater than 0.5 minus a small epsilon, the particle's path is almost surely closed, advancing understanding of stochastic lattice dynamics.

Findings

Particle trajectories are almost surely closed for p > 0.5 - ε.

The study extends knowledge of particle motion in obstructed lattice environments.

Provides probabilistic bounds for particle path closure in Manhattan lattices.

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 $p>\frac{1}{2}-\varepsilon$ with some $\varepsilon>0$.