Question

Consider $n$ individuals {$1,2, \ldots, n$}. For each (unordered) pair of individuals $i \neq j$ we consider a random variable $X_{i,j}$ that can be thought of as the distance between $i$ and $j$. Each individual kills its closest neighbour (everything happens at the same time). Can we say anything about the distribution of the number of survivors in the limit $n \to \infty$?

The case $X_{i,j} = |Y_i-Y_j|$ where $Y_1, \ldots,Y_N$ are $n$ i.i.d random variables uniformly distributed on $[0,1]$ is the famous "birds on wire" problem.

What about the case where the random variables $X_{i,j}$ are independent and exponentially distributed, say? Has it been studied in the literature?

Answer 1

The number of people who kill any given $i$ has asymptotically Poisson distribution (since the events that $j$ kills $i$ for different $j$s are almost independent). Thus the number of survivors $S$ is roughly $N/e$. Since different $i$s are also nearly independent, $S$ is roughly normal.

The oriented version, where dependence is even weaker is just the range of a random function, which is quite simple to analyze.