论文标题
具有无界速度的止动性种系模型的存在和渗透结果
Existence and percolation results for stopped germ-grain models with unbounded velocities
论文作者
论文摘要
我们研究了一般停止种系模型的存在和第一渗透特性。它们是通过由$ \ mathbf {r}^{2} $中的均质平面泊松点过程生成的随机系列定义的。从每个胚芽中，由随机数量的分支组成的谷物生长。每当它的一个分支之一击中另一种谷物时，这种谷物就会停止生长。经典和历史示例是线段模型，谷物是在$ [0,2π）$中以随机速度在随机方向生长的线段模型。在双边线段模型中，各个段都在两个方向上生长。这里考虑了其他示例，例如Brownian模型，其中仅由$ \ Mathbf {r}^{2} $中的独立布朗动作给出了分支。对于无限数量的细菌而存在这种动力学并不明显，我们的第一个结果可确保它处于非常普遍的环境中。特别地，只要随机速度允许订单4的时刻，该线段模型的存在就会证明，该阶段扩展了Daley等人（定理4.3 in \ cite {daley2014two}）的结果，以达到有限速度。我们的结果还涵盖了布朗动态模型。在本文的第二部分中，我们表明具有随机速度的线段模型承认超级指数力矩不会渗透。这改善了最新结果（定理3.2 \ cite {coupier2016absence}）在有限速度的情况下。
We investigate the existence and first percolation properties of general stopped germ-grain models. They are defined via a random set of germs generated by a homogeneous planar Poisson point process in $\mathbf{R}^{2}$. From each germ, a grain, composed by a random number of branches, grows. This grain stops to grow whenever one of its branches hits another grain. The classical and historical example is the line segment model for which the grains are segments growing in a random direction in $ [0,2π)$ with random velocity. In the bilateral line segment model the segments grow in both directions. Other examples are considered here such as the Brownian model where the branches are simply given by independent Brownian motions in $\mathbf{R}^{2}$. The existence of such dynamics for an infinite number of germs is not obvious and our first result ensures it in a very general setting. In particular the existence of the line segment model is proved as soon as the random velocity admits a moment of order 4 which extends the result by Daley et al (Theorem 4.3 in \cite{daley2014two}) for bounded velocity. Our result covers also the Brownian dynamic model. In a second part of the paper, we show that the line segment model with random velocity admitting a super exponential moment does not percolate. This improves a recent result (Theorem 3.2 \cite{coupier2016absence}) in the case of bounded velocity.
