Infinite WARM graphs III, strong reinforcement regime

Abstract

We study the random subgraph $\mathcal E$, consisting of edges reinforced infinitely often, in a reinforcement model on infinite graphs $G$ of bounded degree. The model involves a parameter $\alpha > 0$ governing the strength of reinforcement, and Poisson firing rates $\lambda_v$ at the vertices $v$ of the graph. It was shown earlier that for various graphs $G$, all connected components of $\mathcal E$ are finite when $\alpha \gg 1$ is sufficiently large and that infinite clusters in $\mathcal E$ are possible for suitably chosen $G$ and $\alpha>1$. In this paper, we focus on the finite connected components of $\mathcal E$ in the strong reinforcement regime ($\alpha>1$). When $\alpha$ is sufficiently large, all connected components of $\mathcal E$ are trees. When the firing rates $\lambda_v$ are constant, components are trees of diameter at most 3 when $\alpha$ is sufficiently large. We show that there are infinitely many phase transitions as $\alpha\downarrow 1$. For instance, on the triangular lattice, increasingly large (odd) cycles appear when taking $\alpha\downarrow 1$, while on the square lattice no finite component of $\mathcal E$ contains a cycle for any $\alpha > 1$. Increasingly long paths and other structures appear in both lattices when taking $\alpha\downarrow 1$. In the special case where $G = \mathbb Z$ and $\alpha > 1$, all connected components of $\mathcal E$ are finite and we show that the possible cluster sizes are non-monotone in $\alpha$.