Abstract :

Vertex-Reinforced Random Walk (VRRW), defined by Pemantle (1988), is a random process taking values in the vertex set of a graph G, which is more likely to visit vertices it has visited before. Pemantle and Volkov (1997) considered the case where the underlying graph is the one-dimensionnal integer lattice. 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 one; the proof of this conjecture is the main purpose of this paper.