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arXiv 1908.07375math.PR
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具有随机幂的SIR模型的渗流相变

Percolation phase transitions for the SIR model with random powers

Regine Löffler

备注：
master's thesis at TU Berlin, 2019
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摘要：

本论文考虑三种模型来描述多跳自组织通信系统。这些系统由发送消息的用户组成，这些消息可以跳转到其他用户以到达目标用户。前两个模型已经被广泛研究，而第三个模型是第一次研究。在所有这些模型中，我们的目标是了解在何种条件下，相距遥远的用户可以进行通信。这是连续渗流的基本问题，由Gilbert（1961）引入，他建立了Poisson点过程的布尔模型。在本论文的第一部分中，我们介绍了这个模型，并证明了存在一个相变之间的亚临界阶段，其中没有无限连接的用户组件存在，和超临界阶段，这样的组件存在。我们也考虑随机连接半径的情况，遵循Meester和Roy（1996）。在第二部分中，我们研究了二维或更高维Cox点过程的SINR模型，遵循Tóbiás（2019）。他证明了存在一个相变下一定的稳定性和连通性条件下的强度措施。$\mathbb R^2$中的齐次泊松点过程的SINR模型由Dousse等人（2005）引入。第三部分研究了$\mathbb R^d$、$d \geq 2$中Poisson点过程的随机幂SINR模型的渗流问题。Kong and Yeh（2007）在随机幂的强有界性条件下研究了该模型的$d=2$。在本文中，我们弱化了这些条件，只假设随机幂是非负的、可积的和非a.s.的。零.我们表明，存在一个亚临界相，如果路径损失函数衰减足够快。进一步证明了当随机幂具有足够大的本质上确界和一定的指数矩时，超临界相存在。

英文摘要：

This thesis considers three models which describe a multihop ad-hoc telecommunication system. These systems consist of users sending messages, which can jump to other users to reach the target user. The first two models have already been examined extensively, whereas it is the first time the third model is studied. In all these models our goal is to understand under which conditions users situated far away from each other can communicate. This is the fundamental question of continuum percolation, which was introduced by Gilbert (1961), who established the Boolean model for Poisson point processes. In the first part of this thesis we introduce this model and prove the existence of a phase transition between a subcritical phase, where no infinite connected component of users exists, and a supercritical phase, where such a component exists. We also consider the case of random connection radii, following Meester and Roy (1996). In the second part we study the SINR model for Cox point processes in two or higher dimensions, following Tóbiás (2019). He proved that there exists a phase transition under certain stabilization and connectedness conditions on the intensity measure. The SINR model for homogeneous Poisson point processes in $\mathbb R^2$ was introduced by Dousse et al. (2005). In the third part we study percolation in the SINR model with random powers for Poisson point processes in $\mathbb R^d$, $d \geq 2$. This model was studied by Kong and Yeh (2007) for $d=2$ under strong boundedness conditions on the random power. In this thesis we weaken these conditions and only assume that the random power is nonnegative, integrable and not a.s. zero. We show that there exists a subcritical phase if the path-loss function decays fast enough. Further, we prove that a supercritical phase exists if random power has a large enough essential supremum and some exponential moments.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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