Multi-range percolation on oriented trees: critical curve and limit behavior

We consider an inhomogeneous oriented percolation model introduced by de Lima, Rolla and Valesin. In this model, the underlying graph is an oriented rooted tree in which each vertex points to each of its $d$ children with `short' edges, and in addition, each vertex points to each of its $d^k$ descendant at a fixed distance $k$ with `long' edges. A bond percolation process is then considered on this graph, with the prescription that independently, short edges are open with probability $p$ and long edges are open with probability $q$. We study the behavior of the critical curve $q_c(p)$: we find the first two terms in the expansion of $q_c(p)$ as $k \to \infty$, and prove that the critical curve lies strictly above the critical curve of a related branching process, in the relevant parameter region. We also prove limit theorems for the percolation cluster in the supercritical, subcritical and critical regimes.