Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Percolation phase transitions for the SIR model with random powers   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.

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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ex
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The categories of this paper are: adap-org
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th