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 $p=1/2$ 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 $pgeq 1/2$. 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 [ expleft( - C_1 t^{frac{d}{d+2}}right) leq mathbb{P}_p(tau > t) leq expleft( - 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 $e^{-csqrt{t}}$. Our exponent $d/(d+2)$ also differs from those previously obtained in the case of moving obstacles.