论文标题
在$ \ mathbb {z}^d $上伸展亚临界停车时间的指数衰减
Stretched exponential decay for subcritical parking times on $\mathbb{Z}^d$
论文作者
论文摘要
在$ \ Mathbb {Z}^d $的停车模型中，每个顶点最初被汽车（带有概率$ p $）或空置停车位占据（概率$ 1-P $）。汽车进行独立的随机步行，当他们进入空地时，它们将停在那里，从而使该位置占据。访问被占领地点的汽车只是继续驾驶（继续随机步行）。众所周知，$ p = 1/2 $是一个关键价值，从某种意义上说，原点是A.S.当$ p <1/2 $ $ p <1/2 $时，有限许多不同的汽车访问，当$ p \ geq 1/2 $时，由无限的许多不同的汽车访问。此外，任何给定的汽车A.S.最终以$ p \ leq 1/2 $的公园公园，并以阳性概率为$ p> 1/2 $停车。我们研究了亚临界阶段，并证明了汽车的停车时间$τ$最初是在原点遵守边界\ [\ [\ [\ c_1 t^{\ frac {d exp {d+2} {d+2}} {d+2}}}} \ right） t^{\ frac {d} {d+2}}} \ right）\ for $ p> 0 $足够小。对于$ d = 1 $，我们证明了[0,1/2）$中所有$ p \的这些不平等现象。该结果带来了超临界阶段（$ p> 1/2 $）的不对称性，其中bramson-lebowitz的方法暗示着$ d = 1 $，起源停车位的相应尾巴的相应尾巴是$ e^{ - c \ c \ c \ sqrt {t}} $。我们的指数$ d/（d+2）$与以前在移动障碍的情况下获得的指数也有所不同。
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 $τ$ of the car initially at the origin obeys the bounds \[ \exp\left( - C_1 t^{\frac{d}{d+2}}\right) \leq \mathbb{P}_p(τ> t) \leq \exp\left( - c_2 t^{\frac{d}{d+2}}\right) \] 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.
