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Consider the following model for percolation in an infinite graph: each vertex has a certain region (set of vertices) associated with it, which at the beginning contains only the vertex itself, and grows with time. Each vertex holds a random Poisson clock, all clocks independent, and at each tick of a clock of a vertex $v$, $v$'s region is updated to the current region united with the regions of all neighbors of the $v$ (which might have grown by that time).

The question about this model is whether there exists a vertex whose region percolates = becomes infinite at a finite time with positive probability.

Does this model have a standard name and/or has been studied before?

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This kind of construction comes up in various contexts, for example in analysis of the contact model, or other models for the spread of an infection or a rumour or the like. I don't know if it has a particular name.

For some graphs you could have infinite regions at finite times.

If you impose a condition that, say, the graph has bounded vertex degrees, then the probability that $v$ has an infinite region at any finite time $t$ is zero. For $u$ to be in the region of $v$ at time $t$, there needs to be a path $u=v_0, v_1, \dots, v_n=v$ and times $0 \leq t_1 \leq t_2, \dots, t_n \leq t$ such that the bell at vertex $v_i$ rings at time $t_i$. For any given such path of length $n$, the probability that the bells ring in this way is $O(t^n/n!)$ as $n\to\infty$. If each vertex has degree at most $d$, then the number of such paths of length $n$ ending at $v$ is $O((d-1)^n)$ as $n\to\infty$. Since $\sum (d-1)^n t^n/n!$ is finite, the expected size of $v$'s region at time $t$ is finite.

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