论文标题
在动态远距离渗透方面的接触过程
Contact process on a dynamical long range percolation
论文作者
论文摘要
在本文中，我们在完整的图形$（V，\ Mathcal {e}）$上定义的动态远距离渗透（CPDLP）上介绍了一个联系过程。动态的长距离渗透是在边缘集合$ \ Mathcal {e} $上定义的砍伐过程，该过程将分配给每个边缘的状态。 Edge $ E $的状态以RATE $ v_e $更新，并在更新后用概率$ p_e $开放，否则就关闭了。然后，在此不断发展的随机环境的顶部定义了接触过程，仅使用开放的边缘进行感染，而恢复与背景无关。首先，我们得出的结论是，上层法律存在，并且上层不变性的生存和非平凡性的相变。然后，我们将接触过程与特定感染核进行比较，该核充当下限。因此，我们获得了关键感染率的上限。我们还表明，如果所有边缘的边缘开放率较低，则CPDLP进入免疫阶段，即无论感染率的值如何，它都将无法生存。此外，我们表明，在$ V = \ Mathbb {z} $上，在适当的条件下，在动态远距离渗透的速度上，如果更新速度在任何给定的感染率$λ$中均匀地收敛到零，则CPDLP几乎肯定会消失。
In this paper we introduce a contact process on a dynamical long range percolation (CPDLP) defined on a complete graph $(V,\mathcal{E})$. A dynamical long range percolation is a Feller process defined on the edge set $\mathcal{E}$, which assigns to each edge the state of being open or closed independently. The state of an edge $e$ is updated at rate $v_e$ and is open after the update with probability $p_e$ and closed otherwise. The contact process is then defined on top of this evolving random environment using only open edges for infection while recovery is independent of the background. First, we conclude that an upper invariant law exists and that the phase transitions of survival and non-triviality of the upper invariant coincide. We then formulate a comparison with a contact process with a specific infection kernel which acts as a lower bound. Thus, we obtain an upper bound for the critical infection rate. We also show that if the probability that an edge is open is low for all edges then the CPDLP enters an immunization phase, i.e. it will not survive regardless of the value of the infection rate. Furthermore, we show that on $V=\mathbb{Z}$ and under suitable conditions on the rates of the dynamical long range percolation the CPDLP will almost surely die out if the update speed converges to zero for any given infection rate $λ$.
