Some self-interacting random walks

We will discuss two different classes of self-interacting random walks. Firstly, in joint work with Akira Sakai, we consider a simple model of a random walk with reinforcement but with very short term spatial memory. The simplicity of these 'Senile random walks' enables us to prove many things such as recurrence/transience and an exact expression for the diffusion constant. Secondly, in joint work with Remco van der Hofstad, we derive an expansion for general self-interacting random walks. We use the expansion to prove a central limit theorem for a class of once reinforced random walks with non-zero drift (all dimensions) and for excited random walk (high dimensions), when the reinforcement and excitement parameters are sufficiently small.