  From each point of a Poisson point process start growing a balloon at rate 1.
When two balloons touch, they pop and disappear. Is every point contained in
balloons infinitely often or not? We answer this for the Euclidean space, the
hyperbolic plane and regular trees.
  The result for the Euclidean space relies on a novel 0-1 law for stationary
processes. Towards establishing the results for the hyperbolic plane and
regular trees, we prove an upper bound on the density of any well-separated set
in a regular tree which is a factor of an i.i.d. process.
