Infinite WARM graphs III: Strong reinforcement regime

Abstract

We study a reinforcement process on graphs G of bounded degree. The model involves a parameter α > 0 governing the strength of reinforcement, and Poisson clock rates λ v at the vertices v of the graph. When the Poisson clock at a vertex v rings, one of the edges incident to it is reinforced, with edge e being chosen with probability proportional to its current count (counts start from 1) raised to the power α. The main problem in such models is to describe the (random) subgraph, consisting of edges that are reinforced infinitely often. In this paper, we focus on the finite connected components of in the strong reinforcement regime (α > 1) with clock rates that are uniformly bounded above. We..