In this worksheet, we will consider a natural random walk process, that is not Markovian.
The reinforced random walk on the integers is defined as follows: if you are \(i \in \mathbb{Z}\), then you can only go to \(i+1\) or \(i-1\), and you move to \(i+1\) with probability \(v(i+1)/[v(i-1) + v(i+1)]\), where \(v(j)\) is the total number of times an integer \(j\) has already been visited. Initially, we take \(v=1\) everywhere.
It is a theorem of Tarres (2004) that the walk will eventually get stuck on five vertices. In this worksheet we will try to write some code to illustrate this fact.