Vertex - Reinforced Random Walk on Z Eventually Gets Stuck on Five Points

Vertex-reinforced random walk (VRRW), defined by Pemantle in 1988, is a random process that takes values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov considered the case when the underlying graph is the one-dimensional integer lattice Z. They proved that the range is almost surely finite and that with positive probability the range contains exactly five points. They conjectured that this second event holds with probability 1. The proof of this conjecture is the main purpose of this paper.