April 2023 Parking on the integers
Michał Przykucki, Alexander Roberts, Alex Scott
Author Affiliations +
Ann. Appl. Probab. 33(2): 1076-1101 (April 2023). DOI: 10.1214/22-AAP1836


Models of parking in which cars are placed randomly and then move according to a deterministic rule have been studied since the work of Konheim and Weiss in the 1960s. Recently, Damron, Gravner, Junge, Lyu, and Sivakoff ((2019) Ann. Appl. Probab. 29 2089–2113) introduced a model in which cars are both placed and move at random. Independently at each point of a Cayley graph G, we place a car with probability p, and otherwise an empty parking space. Each car independently executes a random walk until it finds an empty space in which to park. In this paper we introduce three new techniques for studying the model, namely the space-based parking model, and the strategies for parking and for car removal. These allow us to study the original model by coupling it with models where parking behaviour is easier to control. Applying our methods to the one-dimensional parking problem in Z, we improve on previous work, showing that for p<1/2 the expected journey length of a car is finite, and for p=1/2 the expected journey length by time t grows like t3/4 up to a polylogarithmic factor.

Funding Statement

The first author was supported by the EPSRC grant EP/P026729/1.
The third author was supported by a Leverhulme Trust Research Fellowship.


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Michał Przykucki. Alexander Roberts. Alex Scott. "Parking on the integers." Ann. Appl. Probab. 33 (2) 1076 - 1101, April 2023. https://doi.org/10.1214/22-AAP1836


Received: 1 August 2019; Revised: 1 February 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 07692284
MathSciNet: MR4564421
Digital Object Identifier: 10.1214/22-AAP1836

Primary: 60K35 , 82C22
Secondary: 82B26

Keywords: Cayley graphs , parking problems , Random walks

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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