The existence of the most recent common maternal ancestor, the mitochondrial Eve, is studied using results from the Galton-Watson process. To accomplish this, ancestral trees are generated...Show moreThe existence of the most recent common maternal ancestor, the mitochondrial Eve, is studied using results from the Galton-Watson process. To accomplish this, ancestral trees are generated according to certain assumptions and progeny distributions. Results from the Galton-Watson process are first recreated from which the condition for an infinite tree is derived. These results are applied to a model of the mitochondrial Eve. Assuming a number of women contemporary to the mitochondrial Eve, the maximum probability of one lineage surviving is also determined in a specific situation.Show less
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...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
In this thesis we study a class of self-interacting random walk. The specific behaviour, which can differ from a simple random walk, is presented in the form of simulations and depends on some...Show moreIn this thesis we study a class of self-interacting random walk. The specific behaviour, which can differ from a simple random walk, is presented in the form of simulations and depends on some chosen parameters. The model for the random walk is as follows: the walker chooses a starting position on the integers (1-dimensional walk) and an initial value/weight is given to every edge between two neighbouring sites called the local time profile. Our stochastic model represents the probability to jump one position, after one unit of time, to the right which also depends on the local time profile. When the walker doesn’t jump right the walker jumps one position to the left. The local time of the corresponding edge is raised with an amount of one and the local time profile is updated after every step. The underlying idea in this model is the possibility to choose parameters in such a way the local time profile determines the qualitative asymptotic behaviour of the walk. We start with the just described model and prove for a range of parameters that the walk eventually gets stuck on a single edge. This will be done in three steps. First we prove the walk gets stuck with positive probability. Next we show the walk must have a finite range and we use Rubin’s Theorem to complete the proof. Further we investigate extended models where the walker can jump one or two positions at a time and find out that for one choice of parameters the type of behaviour is not unique. In the remainder we discuss the main result from the article Stuck walks by Erschler, T´oth and Werner. The theorem stated there gives a range of parameters such that the walk gets stuck on two, three, . . . edges.Show less
