Abstract
Place a car independently with probability $p$ at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when $p\geq1/2$, and only finitely many times otherwise.
Citation
Michael Damron. Janko Gravner. Matthew Junge. Hanbaek Lyu. David Sivakoff. "Parking on transitive unimodular graphs." Ann. Appl. Probab. 29 (4) 2089 - 2113, August 2019. https://doi.org/10.1214/18-AAP1443
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