论文标题
连续时间青蛙模型的爆炸和非探索
Explosion and non-explosion for the continuous-time frog model
论文作者
论文摘要
我们考虑$ \ mathbb {z} $上的连续时间青蛙模型。在时间$ t = 0 $时，$ x \ in \ mathbb {z} $有$η（x）$粒子，每个粒子由随机变量表示。特别是，$（η（x））_ {x \ in \ mathbb {z}} $是具有共同分布的独立随机变量的集合，$μ$，$μ（\ mathbb {z} _+）= 1 $。原点上的颗粒是活跃的，所有其他被假定为休眠或睡觉的颗粒。活动粒子在$ \ mathbb {z} $中执行一个简单的对称连续时间随机步行（即，与$ \ exp（1）$ - 分布式跳跃时间和$ -1 $和$ 1 $的随机步行，每个概率$ 1/2 $），独立于所有其他粒子。睡眠颗粒保持静止，直到活性粒子首次到达其位置为止。到达后，它们变得活跃并开始自己的简单随机步行。给出了一组条件，以确保连续时间青蛙模型的爆炸分别爆炸，分别是非探索。我们的结果特别表明，如果$μ$是$ e^{y \ ln y} $的分布，则具有非负随机变量$ y $满足$ \ mathbb {e} y <\ infty $，然后是A.S.没有爆炸发生。另一方面，如果$ a \ in（0,1）$和$μ$是$ e^x $的分布，其中$ \ mathbb {p} \ {x \ geq t \} = t \} = t^{ - a} $，$ t \ geq 1 $，然后爆炸发生。证明依赖于某种类型的渗滤模型的比较，我们称之为完全不对称的离散不均匀的布尔渗透。
We consider the continuous-time frog model on $\mathbb{Z}$. At time $t = 0$, there are $η(x)$ particles at $x\in \mathbb{Z}$, each of which is represented by a random variable. In particular, $(η(x))_{x \in \mathbb{Z} }$ is a collection of independent random variables with a common distribution $μ$, $μ(\mathbb{Z}_+) = 1$. The particles at the origin are active, all other ones being assumed as dormant, or sleeping. Active particles perform a simple symmetric continuous-time random walk in $\mathbb{Z} $ (that is, a random walk with $\exp(1)$-distributed jump times and jumps $-1$ and $1$, each with probability $1/2$), independently of all other particles. Sleeping particles stay still until the first arrival of an active particle to their location; upon arrival they become active and start their own simple random walks. Different sets of conditions are given ensuring explosion, respectively non-explosion, of the continuous-time frog model. Our results show in particular that if $μ$ is the distribution of $e^{Y \ln Y}$ with a non-negative random variable $Y$ satisfying $\mathbb{E} Y < \infty$, then a.s. no explosion occurs. On the other hand, if $a \in (0,1)$ and $μ$ is the distribution of $e^X$, where $\mathbb{P} \{X \geq t \} = t^{-a}$, $t \geq 1$, then explosion occurs a.s. The proof relies on a certain type of comparison to a percolation model which we call totally asymmetric discrete inhomogeneous Boolean percolation.
