Geometric properties of Poisson matchings

In this thesis we discuss Poisson matchings, which can be seen as ’random graphs’ with an infinite set of random vertices U ⊂ R. The set ’U’ is the set of ’arrival times’ in a Poisson process. It is our goal to gain more insight in a specific matching type called the ’stable multi-matching’, which has some extra nice properties. We are mainly concerned with the question whether infinite components exist in these random graphs . In this paper we will study this question for the stable multi-matching with a specific degree distribution. Showing the existence of an infinite component in this matching is still an open question, but an overwhelming amount of simulation results seems to suggest a positive answer. We will also add our simulation results in support of the conjecture that such an infinite component does exist (with probability 1). We will mainly use and study the work of Alexander Holroyd and for many results will refer to his papers on this subject. Also, we try to give the...Show moreIn this thesis we discuss Poisson matchings, which can be seen as ’random graphs’ with an infinite set of random vertices U ⊂ R. The set ’U’ is the set of ’arrival times’ in a Poisson process. It is our goal to gain more insight in a specific matching type called the ’stable multi-matching’, which has some extra nice properties. We are mainly concerned with the question whether infinite components exist in these random graphs . In this paper we will study this question for the stable multi-matching with a specific degree distribution. Showing the existence of an infinite component in this matching is still an open question, but an overwhelming amount of simulation results seems to suggest a positive answer. We will also add our simulation results in support of the conjecture that such an infinite component does exist (with probability 1). We will mainly use and study the work of Alexander Holroyd and for many results will refer to his papers on this subject. Also, we try to give the reader some impression of the arguments involved in proving statements about geometric properties of the Poisson matchings. Intuitively some of these problems can seem misleadingly simple, but often there’s many subtleties and difficult mathematics involved in proving statements about geometric properties. So, don’t be misleaded by easy questions with difficult answers!Show less

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Geometric properties of Poisson matchings