Percolation phase transitions for the SIR model with random powers

Abstract

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 T\'obi\'as (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.