Self-Interacting Random Walks and Related Braching-Like Processes

In this thesis we study two different types of self-interacting random walks. First, we study excited random walk in a deterministic, identically-piled cookie environment under the constraint that the total drift contained in the cookies at each site is finite. We show that the walk is recurrent when this parameter is between -1 and 1 and transient when it is less than -1 or greater than 1. In the critical case, we show that the walk is recurrent under a mild assumption on the environment. We also construct an environment where the total drift per site is 1 but in which the walk is transient. This behavior was not present in previously-studied excited random walk models.

Second, we study the "have your cookie and eat it'' random walk proposed by Pinsky, who already proved criteria for determining when the walk is recurrent or transient and when it is ballistic. We establish limiting distributions for both the hitting times and position of the walk in the transient regime which, depending on the environment, can be either stable or Gaussian.