Given $\rho$ particles uniformly distributed on a plane within a unit square ($\rho > 1$), each particle has another particle that is closest to it; the median of those nearest distances is called $\tilde{d}$. Here's a graph where each point represents a random distribution of particles:
Taking the mean $\tilde{d}$ for each value of $\rho$ produces a nice graph which aligns with this function:
$\tilde{d} = \frac{k_1}{\rho}+\frac{k_2}{\sqrt{\rho}}$
The constants are estimated here to be:
$k_1 \approx 0.1705$
$k_2 \approx 0.4649$
These constants have been calculated empirically (interpolating the empirical points to a function); is there a way to calculate them theoretically? In other words, is there a purely mathematical proof of a relationship between $\rho$ and $\tilde{d}$?
EDIT: I managed to mathematically derive $\tilde{d}$ for $\rho = 2$; but this is by far the easiest and it's already a long formula:
