Infinite WARM graphs III: strong reinforcement regime

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 $\mathcal{E}_\infty$, consisting of edges that are reinforced infinitely often. In this paper, we focus on the finite connected components of $\mathcal{E}_\infty$ in the strong reinforcement regime (α > 1) with clock rates that are uniformly bounded above. We show here that when α is sufficiently large, all connected components of $\mathcal{E}_\infty$ are trees. When the firing rates λ_v are constant, we show that all components are trees of diameter at most 3 when α is sufficiently large, and that there are infinitely many phase transitions as $\alpha\downarrow 1$. For example, on the triangular lattice, increasingly large (odd) cycles appear as $\alpha\downarrow 1$ (while on the square lattice no finite component of $\mathcal{E}_\infty$ contains a cycle for any α > 1). Increasingly long paths and other structures appear in both lattices when taking $\alpha\downarrow 1$. In the special case where $G = \mathbb{Z}$ and α > 1, all connected components of $\mathcal{E}_\infty$ are finite and we show that the possible cluster sizes are non-monotone in α. We also present several open problems.