2 problems

Let $\mathcal{H}$ be a non-amenable (quasi)transitive graph, with percolation threshold $p_{\rm c}(\mathcal{H})$ and uniqueness threshold $p_{\rm u}(\mathcal{H})$. For a sequence o…

Consider the branching random walk from the preceding example, with particles on $\mathbb{Z}$ having two children at $n+1$ with probability $p-\varepsilon$, one child at $n-1$ with…