Sharpness and critical scaling of parking
Abstract
In the parking model, each site of the d-dimensional lattice independently starts with one car with probability p or one parking spot with probability 1−p. Cars move according to independent discrete-time simple random walks and park at the first spot they find free. We prove that in the critical regime p=1/2, the expected number of visits to a site in n rounds is of order n(4−d)/4 for d≤3 and log n for d≥4. We also prove that in the subcritical regime p∈(0,1/2), the parking-time tail is bounded above and below by stretched exponentials with exponent d/(d+2). As p↑1/2, we also determine the divergence of the expected total number of visits to a site: its order is (1−2p)−3, (1−2p)−1 and (1−2p)−1/3 in dimensions one, two and three, respectively, and log(1/(1−2p)) in dimensions four and higher. Our proof uses a representation of the parking process as the divisible sandpile of Levine and Peres plus a martingale-type term. These results answer questions posed by Damron, Gravner, Junge, Lyu and Sivakoff (2019).
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BibTeX
@misc{bourabee2026sharpness,
title = {{Sharpness and critical scaling of parking}},
author = {Ahmed Bou-Rabee and Christoforos Panagiotis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02820},
eprint = {2609.02820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02820},
}
RIS
TY - UNPB
TI - Sharpness and critical scaling of parking
AU - Ahmed Bou-Rabee
AU - Christoforos Panagiotis
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02820
UR - https://arxiv.org/abs/2609.02820
AB - In the parking model, each site of the d-dimensional lattice independently starts with one car with probability p or one parking spot with probability 1−p. Cars move according to independent discrete-time simple random walks and park at the first spot they find free. We prove that in the critical regime p=1/2, the expected number of visits to a site in n rounds is of order n^(4−d)/4 for d≤3 and log n for d≥4. We also prove that in the subcritical regime p∈(0,1/2), the parking-time tail is bounded above and below by stretched exponentials with exponent d/(d+2). As p↑1/2, we also determine the divergence of the expected total number of visits to a site: its order is (1−2p)^−3, (1−2p)^−1 and (1−2p)^−1/3 in dimensions one, two and three, respectively, and log(1/(1−2p)) in dimensions four and higher. Our proof uses a representation of the parking process as the divisible sandpile of Levine and Peres plus a martingale-type term. These results answer questions posed by Damron, Gravner, Junge, Lyu and Sivakoff (2019).
ER -