تقرير
Stretched exponential decay for subcritical parking times on $\mathbb{Z}^d$
العنوان: | Stretched exponential decay for subcritical parking times on $\mathbb{Z}^d$ |
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المؤلفون: | Damron, Michael, Lyu, Hanbaek, Sivakoff, David |
سنة النشر: | 2020 |
المجموعة: | Mathematics |
مصطلحات موضوعية: | Mathematics - Probability, 60K35 |
الوصف: | 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. Comment: 10 pages |
نوع الوثيقة: | Working Paper |
URL الوصول: | http://arxiv.org/abs/2008.05072 |
رقم الأكسشن: | edsarx.2008.05072 |
قاعدة البيانات: | arXiv |
الوصف غير متاح. |