# Parking on the integers

## TL;DR

This paper introduces new techniques to analyze a random parking model on the integers, showing finite expected journey lengths for low car densities and a specific growth rate at the critical density.

## Contribution

The authors develop three innovative methods, including space-based models and parking strategies, to better understand and analyze the random parking process on Cayley graphs.

## Key findings

- Expected journey length is finite for p<1/2.
- At p=1/2, journey length grows like t^{3/4} up to polylogarithmic factors.
- New techniques improve understanding of parking behavior on the integers.

## 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 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 $\mathbb{Z}$, 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.