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

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 $p=1\mathrm{/}2$ is a critical value in the sense that the origin is a.s. visited by finitely many distinct cars when $p<1\mathrm{/}2$, and by infinitely many distinct cars when $p\ge 1\mathrm{/}2$. Furthermore, any given car a.s. eventually parks for $p \leq 1/2$andwithpositiveprobabilitydoesnotparkfor$p > 1/2$. 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 $$\exp (-C_1{t}^{\frac{d}{d+2}})\le {\mathbb{P}}_p(\tau >t)\le \exp (-c_2{t}^{\frac{d}{d+2}})$$ for $p>0$ sufficiently small. For $d=1$, we prove these inequalities for all $p\in [0,1\mathrm{/}2)$. This result presents an asymmetry with the supercritical phase ($p>1\mathrm{/}2$), 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 $d\mathrm{/}(d+2)$ also differs from those previously obtained in the case of moving obstacles.