The Balloon Process
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What is this?
We start with a random collection of points, either in the Euclidean plane (first picture) or in the Poincaré hyperbolic disk (second picture), chosen according to a Poisson point process. The points are thought of as the centers of balloons, which are initially deflated. The balloons are then inflated at a constant unit rate, so that they all have radius t at time t. As time passes, some balloons will inevitably collide with one another. Whenever two balloons collide, they both pop, and the points corresponding to their centers are removed. This describes a process of an evolving collection of points, which becomes smaller and smaller with time. The picture depicts the balloons which have not yet been popped at a certain time in blue, and balloons which have already popped by this time in gray (with their size being as when they popped). The lines connect centers of balloons which collided with each other.
Mathematics
As the balloon process evolves, there will times when the origin is covered by a balloon, and there will be times when it is not. Will the set of times when the origin is covered be bounded or unbounded? As it turns out, in the Euclidean plane, this set of times is always unbounded. This is true regardless of the initial density of points. In fact, the same is also true in Euclidean space of any dimension. On the other hand, in the hyperbolic disk, this set of times is always bounded. This too is true regardless of the initial density of points. These facts were proved in [1].
What causes this difference in behavior between the Euclidean plane and the hyperbolic disk?
The reason has to do with the very different geometry of the two spaces. A primary such difference is that in Euclidean space, the area of a ball of radius t grows polynomially fast in t, whereas in the hyperbolic disk, it grows exponentially fast in t.
The balloon process has connections with stable matchings, first studied by Gale and Shapley [2]. By pairing the centers of balloons which collide with one another (depicted by lines in the pictures above), one obtains a stable matching of the intial points of the process.
Stable matchings of Poisson point processes in Euclidean space were studied in [3,4]. Actually, it is not obvious that the balloon process is well defined, and the fact that it is so is a consequence of the results on stable matchings.
References
[1] O. Angel, G. Ray and Y. Spinka. A tale of two balloons. 2021
[2] D. Gale and L. S. Shapley. College admissions and the stability of marriage. The American Mathematical Monthly, 1962
[3] O. Häggström and R. Meester. Nearest neighbor and hard sphere models in continuum
percolation. Random Structures & Algorithms, 1996
[4] A. Holroyd, R. Pemantle, Y. Peres and O. Schramm. Poisson matching. Annales de l'Institut Henri Poincaré, 2009