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This thesis considers three models which describe a multihop ad-hoctelecommunication system. These systems consist of users sending messages,which can jump to other users to reach the target user. The first two modelshave already been examined extensively, whereas it is the first time the thirdmodel is studied. In all these models our goal is to understand under whichconditions users situated far away from each other can communicate. This is thefundamental question of continuum percolation, which was introduced by Gilbert(1961), who established the Boolean model for Poisson point processes. In thefirst part of this thesis we introduce this model and prove the existence of aphase transition between a subcritical phase, where no infinite connectedcomponent of users exists, and a supercritical phase, where such a componentexists. We also consider the case of random connection radii, following Meesterand Roy (1996). In the second part we study the SINR model for Cox pointprocesses in two or higher dimensions, following T\'obi\'as (2019). He provedthat there exists a phase transition under certain stabilization andconnectedness conditions on the intensity measure. The SINR model forhomogeneous Poisson point processes in $\mathbb R^2$ was introduced by Dousseet al. (2005). In the third part we study percolation in the SINR model withrandom powers for Poisson point processes in $\mathbb R^d$, $d \geq 2$. Thismodel was studied by Kong and Yeh (2007) for $d=2$ under strong boundednessconditions on the random power. In this thesis we weaken these conditions andonly 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 decaysfast enough. Further, we prove that a supercritical phase exists if randompower has a large enough essential supremum and some exponential moments.

Percolation phase transitions for the SIR model with random powers