Let $T$ be a regular rooted tree. For every natural number $n$, let $B_n$ bethe 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 isgoverned by two opposing mechanisms: Vertices become occupied at rate $1$,independently for all vertices. Independently thereof and independently for allvertices, "lightning" hits vertices at rate $lambda(n) > 0$. When a vertex ishit by lightning, its occupied cluster instantaneously becomes vacant. Now suppose that $lambda(n)$ decays exponentially in $n$ but much moreslowly than $1/|B_n|$. We show that then there exist a supercritical time$au$ and $epsilon > 0$ such that the forest-fire model on $B_n$ between time$0$ and time $au + epsilon$ tends to the following process on $T$ as $n$goes to infinity: At time $0$ all vertices are vacant. Between time $0$ andtime $au$ vertices become occupied at rate $1$, independently for allvertices. At time $au$ all infinite occupied clusters become vacant. Betweentime $au$ and time $au + epsilon$ vertices again become occupied at rate$1$, independently for all vertices. At time $au + epsilon$ all occupiedclusters are finite. This process is a dynamic version of self-destructivepercolation.
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机译：令$ T $为规则的有根树。对于每个自然数$ n $，令$ B_n $为顶点的有限子树，图与根的距离最大为$ n $。请考虑在$ B_n $上使用以下林火模型：每个顶点可以为“ vacant”或“占据”。在时间$ 0 $，所有顶点均空着。然后，该过程由两个相反的机制控制：顶点以$ 1 $的速率被占用，对于所有顶点都是独立的。独立于所有顶点，并且独立于所有顶点，“闪电”以$ lambda（n）> 0 $的速率命中顶点。当一个顶点被闪电击中时，它所占据的簇瞬间变空。现在假设$ lambda（n）$在$ n $中呈指数衰减，但比$ 1 / | B_n | $慢得多。我们表明，然后存在一个超临界时间$ tau $和$ epsilon> 0 $，使得在时间$ 0 $和时间$ tau + epsilon $之间的$ B_n $上的森林火灾模型趋向于以下过程$ T $作为$ n $进入无穷大：在时间$ 0 $处，所有顶点都是空的。在时间$ 0 $和时间$ tau $顶点之间，速率为$ 1 $，对于所有顶点而言都是独立的。在$ tau $时，所有无限占据的群集都变为空置。在时间$ tau $和时间$ tau + epsilon $顶点之间再次以速率$ 1 $被占用，对于所有顶点都是独立的。在时间$ tau + epsilon $时，所有占用的簇都是有限的。此过程是自我破坏渗透的动态版本。
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