Self-destructive percolation as a limit of forest-fire models on regular rooted trees

Let $T$ be a regular rooted tree. For every natural number $n$, let $B_n$ be the finite subtree of vertices with graph distance at most $n$ from the root. Consider the following forest-fire model on $B_n$: Each vertex can be "vacant" or "occupied". At time $0$ all vertices are vacant. Then the process is governed by two opposing mechanisms: Vertices become occupied at rate $1$, independently for all vertices. Independently thereof and independently for all vertices, "lightning" hits vertices at rate $λ(n) > 0$. When a vertex is hit by lightning, its occupied cluster instantaneously becomes vacant. Now suppose that $λ(n)$ decays exponentially in $n$ but much more slowly than $1/|B_n|$. We show that then there exist a supercritical time $τ$ and $ε> 0$ such that the forest-fire model on $B_n$ between time $0$ and time $τ+ ε$ tends to the following process on $T$ as $n$ goes to infinity: At time $0$ all vertices are vacant. Between time $0$ and time $τ$ vertices become occupied at rate $1$, independently for all vertices. At time $τ$ all infinite occupied clusters become vacant. Between time $τ$ and time $τ+ ε$ vertices again become occupied at rate $1$, independently for all vertices. At time $τ+ ε$ all occupied clusters are finite. This process is a dynamic version of self-destructive percolation.

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