This is a follow up to this question where the following problem is explored (for $D=2$):

$n$ riflemen are distributed at random points in $[0,1]^D$. At a signal, each one shoots at and kills his nearest neighbor. What is the expected number $E_D[n]$ of riflemen who are left alive?

I explored the large $D$ limit numerically in this answer which shows that $\frac{E_D[n]}{n}$ is an increasing function of $D$ (for any fixed $n$) which seems to be converging to some limit $f_n = \lim\limits_{D\to\infty}\frac{E_D[n]}{n} < 1$ and $f_n$ also seems to be an increasing function of $n$:


If the box is periodic (making it a torus) we have the simple result $\lim\limits_{n,D\to\infty} \frac{E_D[n]}{n} = \frac{1}{e}$.

My question is the following; what is the limit: $$\lim\limits_{n\to\infty} f_n = \lim\limits_{n\to\infty}\lim\limits_{D\to\infty} \frac{E_D[n]}{n} = ~~?$$ i.e. what does the survivor fraction approch as $n,D\to\infty$?

I suspect it might be unity and the reason I belive this is because the corner effects (i.e. a player in a corner is less likely to be shot than a player in the center) become more and more prominent as we increase $D$. This can be seen by comparing with the case where we make the box periodic in which the results differ greatly for large $D$ (see the figure above). If we consider a $D$-sphere at the center of the box with radius $r=\frac{1}{2}$ then the volume of this sphere goes to zero as $D\to \infty$ so we would expect most particles to be in corner-regions for large $D$. The distance from a corner to another corner is always larger than the distance from a corner to the center. Thus in the some large $n,D$ limit I would roughly expect a typical situation to be something like one particle in each corner all shooting at the person being closest to the center giving us a survivor fraction that would be close to $1$.

I'm unable to solve this so therefore I'm asking it here. I'm also interested in heuristical arguments that can shed some more light on this problem.

Some relevant papers in the litterature analyzing the rifleman problem (for more see references within the papers below):

  1. "The Rifle-Problem" R. Abilock; American Monthly; (1967)
  2. "Vicious neighbor problem" R.Tao and F.Y.Wu; J. Phys. A: Math. Gen. 20 (1987)
  3. "Nearest-Neighbor Graphs" S.R.Finch; Unpublished note; (2008)


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