Suppose we are given an infinite sequence $(\mathbf{r}_n)_{n=1}^\infty$ of 2D points, spread uniformly over space, with a given density $\rho$.
What is the supremum of the percentage of surface covered by the circles with centers $\mathbf{r}_n$ and radii $a_n$, such that the circles do not overlap?
The answer will obviously not depend on the sequence $(\mathbf{r}_n)_{n=1}^\infty$. I also think that $\rho$ will just become a multiplicative constant.
Clarification:
$(\mathbf{r}_n)_{n=1}^\infty$ is a fixed sequence of 2D points distributed uniformly (constant density) over the 2 dimensional plane. Not all sequences of radii $(a_n)_{n=1}^\infty$ are such that the circles do not overlap, but some do. Therefore, the supremum over this set of the percentage of the plane covered by the circles exists. My question is whether this percentage can be calculated or how to approach this problem.
An image is worth a thousand words:
Since the 2D surface is infinitely large, no probability theory is required and the solution will not depend on $(\mathbf{r}_n)_{n=1}^\infty$.
Problem statement as a limit, ($\rho=1$ for simplicity):
Given $N$ uniformly distributed random 2D vectors on a square of size $\sqrt{N}\times\sqrt{N}$, called $\mathbf{r}_1,\cdots,\mathbf{r}_N$. Define:
$$f(\mathbf{r}_1,\cdots,\mathbf{r}_N)=\sup_{a_1,\cdots,a_N}\left\{\frac{1}{\underbrace{N}_{\text{total area}}}\underbrace{\sum_{i=1}^N\pi a_i^2}_{\text{filled area}}:\underbrace{\forall i\forall j\neq i:\|\mathbf{r}_i-\mathbf{r}_j\|\leq a_i+a_j}_{\text{no circles overlap}}\right\}$$
What is $\lim_{N\to\infty}f(\mathbf{r}_1,\cdots,\mathbf{r}_N)$?
I hope everybody agrees that this number will not depend on the choice of $(\mathbf{r}_n)_{n=1}^\infty$, since the 2D surface is infinitely large, but I am afraid my question is ill-posed (does taking the $\mathbb{E}$ help?).