# Does the following percolation model have a name?

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?

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.