A nice random-cleaning-walk through a square

Question

Consider a square of side $L$. Create inside the square $N$ points randomly distributed. Starting from the center of the square (probably not important where you start) move straight to the closest point and erase it. Keep going with the same logic until the square is clean. What is the length $D$ of the polygonal chain you've covered? $$D = f(L,N)?$$ I've divided the square's area $A$ in $N$ equilateral triangles of area $a = A/N = L^2/N$. I've made this hypothesis: being $w$ the triangles' apothem and $d$ the length of the single step of the chain $d = 2 w$. Using the relation beetween the apothem and the area in an equilateral triangle I've expressed $d$ as a function of $L$ and $N$: $$d=2w=\frac{2}{\sqrt{3\sqrt{3}}}\frac{L}{\sqrt{N}}$$ So with the hypothesis $D = N d$ I've found a candidate for the relation I was looking for: $$D = N d = N \frac{2}{\sqrt{3\sqrt{3}}}\frac{L}{\sqrt{N}}$$ After some manupulation $$D = \frac{2}{\sqrt{3\sqrt{3}}} L \sqrt{N}\simeq 0.877 L \sqrt{N}$$ I've set up some Montecarlo simulations and the results seem consistant. What do you think about it? Could you spot any flaws in my reasoning? Or better, cold you find a way to prove it's right?