Phase Transition in Vertex-Reinforced Random Walks on $${\mathbb{Z}}$$ with Non-linear Reinforcement

Abstract

Vertex-reinforced random walk is a random process which visits a site with probability proportional to the weight w k of the number k of previous visits. We show that if w k ∼ k α, then there is a large time T 0 such that after T 0 the walk visits 2, 5, or ∞ sites when α 1, respectively. More general results are also proven.