Parking on the integers
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 bo...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
Institute of Mathematical Statistics
2023
|
| _version_ | 1797109238465560576 |
|---|---|
| author | Przykucki, M Roberts, A Scott, A |
| author_facet | Przykucki, M Roberts, A Scott, A |
| author_sort | Przykucki, M |
| collection | OXFORD |
| description | 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
t
3
/
4
up to a polylogarithmic factor. |
| first_indexed | 2024-03-07T07:38:59Z |
| format | Journal article |
| id | oxford-uuid:2ce8dc71-8ff9-4eed-9082-de0bf51b10af |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:38:59Z |
| publishDate | 2023 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | oxford-uuid:2ce8dc71-8ff9-4eed-9082-de0bf51b10af2023-04-06T12:04:03ZParking on the integersJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:2ce8dc71-8ff9-4eed-9082-de0bf51b10afEnglishSymplectic ElementsInstitute of Mathematical Statistics2023Przykucki, MRoberts, AScott, AModels 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 t 3 / 4 up to a polylogarithmic factor. |
| spellingShingle | Przykucki, M Roberts, A Scott, A Parking on the integers |
| title | Parking on the integers |
| title_full | Parking on the integers |
| title_fullStr | Parking on the integers |
| title_full_unstemmed | Parking on the integers |
| title_short | Parking on the integers |
| title_sort | parking on the integers |
| work_keys_str_mv | AT przykuckim parkingontheintegers AT robertsa parkingontheintegers AT scotta parkingontheintegers |