A square lamina contains $n$ independent uniformly random points. At a given time, each point becomes the centre of a disc whose radius grows from $0$, at say $1$ cm per second, and stops growing when the disc hits another disc.
Consider the discs after all growth has stopped. Here is an example with $n=20$.
Let $P=$ proportion of the square that is covered (i.e. occupied) by the discs. In the example above, $P\approx0.383$.
What does $P$ approach as $n\to\infty$ ?
The shape of the lamina (for example, square) does not matter, since we are taking $n\to\infty$. The rate of growth (for example, $1$ cm per second) does not matter, as long as all the discs start growing at the same time and grow at the same rate.
My thoughts
I tried to find the probability that a new random point in the square lies in one of the existing discs. I also tried to find the average area of a disc. But I haven't succeeded. These questions seem to be complicated by the fact that the size of a point's disc is determined not only by the point's distance to its neighbors, but also its neighbors' distances to their neighbors, and so on.
Context
This question was inspired by another question, "A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?". Both of these questions are about inherent properties of the $2D$ Poisson process.