Stretched exponential decay for subcritical parking times on $\mathbb{Z}^d$

TL;DR

This paper investigates the tail behavior of parking times in a lattice parking model, establishing stretched exponential decay bounds in the subcritical phase for small occupation probabilities, with specific results for one-dimensional cases.

Contribution

It provides the first rigorous bounds on the tail decay of parking times in the subcritical phase, revealing a stretched exponential decay with an explicit exponent depending on the dimension.

Findings

Tail of parking time decays as a stretched exponential with exponent d/(d+2)

Results are valid for small p in all dimensions and for all p in [0,1/2) when d=1

Contrasts with supercritical phase where decay is like e^{-c√t}

Abstract

In the parking model on $\mathbb{Z}^d$, each vertex is initially occupied by a car (with probability $p$) or by a vacant parking spot (with probability $1-p$). Cars perform independent random walks and when they enter a vacant spot, they park there, thereby rendering the spot occupied. Cars visiting occupied spots simply keep driving (continuing their random walk). It is known that $p=1/2$ is a critical value in the sense that the origin is a.s. visited by finitely many distinct cars when $p<1/2$, and by infinitely many distinct cars when $p\geq 1/2$. Furthermore, any given car a.s. eventually parks for $p \leq 1/2$ and with positive probability does not park for $p > 1/2$. We study the subcritical phase and prove that the tail of the parking time $\tau$ of the car initially at the origin obeys the bounds \[ \exp\left( - C_1 t^{\frac{d}{d+2}}\right) \leq \mathbb{P}_p(\tau > t) \leq \exp\left( - c_2 t^{\frac{d}{d+2}}\right) \] for $p>0$ sufficiently small. For $d=1$, we prove these inequalities for all $p \in [0,1/2)$. This result presents an asymmetry with the supercritical phase ($p>1/2$), where methods of Bramson--Lebowitz imply that for $d=1$ the corresponding tail of the parking time of the parking spot of the origin decays like $e^{-c\sqrt{t}}$. Our exponent $d/(d+2)$ also differs from those previously obtained in the case of moving obstacles.