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 $\alpha$.