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

Abstract

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.

Citation

Download Citation

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

Information

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

Subjects:
Primary: 60K35 , 82C22
Secondary: 82B26

Keywords: Cayley graphs , parking problems , Random walks

Rights: Copyright © 2023 Institute of Mathematical Statistics

JOURNAL ARTICLE
26 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

Vol.33 • No. 2 • April 2023
Back to Top