Self-Interacting Random Walk

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 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...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

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Self-Interacting Random Walk