Strongly vertex-reinforced jump process on graph with bounded degree

Abstract

We study asymptotic behaviours of a non-linear vertex-reinforced jump process defined on an arbitrary infinite graph with bounded degree. We prove that if the reinforcement function \(w\) is reciprocally integrable and strictly increasing, then the process visits only a finite number of vertices. In the case where \(w\) is asymptotically equal to a super-linear polynomial, we show that the process eventually gets stuck on a star-shaped subgraph and there is exactly one vertex with unbounded local time.