A (dis)continuous percolation phase transition on the hierarchical lattice

ESB 4127

For long-range percolation on $\mathbb{Z}$ with translation-invariant edge kernel $J$, it is a classical theorem of Aizenman and Newman (1986) that the phase transition is discontinuous when $J(x-y)$ is of order $|x-y|^{-2}$ and that there is no phase transition at all when $J(x-y)=o(|x-y|^{-2})$. We prove analogous theorems for the hierarchical lattice, where the relevant threshold is at $|x-y|^{-2d} \log \log |x-y|$ rather than $|x-y|^{-2}$: There is a continuous phase transition for kernels of larger order, a discontinuous phase transition for kernels of exactly this order, and no phase transition at all for kernels of smaller order. Based on joint work with Tom Hutchcroft.