Average minimum distance between $n$ points generate i.i.d. with uniform dist.

Question

Let $U$ be distributed uniformly on $[0,a]$. Now suppose we generate $n$ independent points according to $U$.

What is the average minimum distance between these $n$ points? That is \begin{align} E\left[ \min_{i,j\in [1,n]: i\neq j} |U_i-U_j|\right] \end{align}

Is this a correct formulation of average minimum distance between?

Answer 1

Assume $a=1$ for simplicity. Let $M$ be the minimum distance among these $n$ points. Note $M$ is always at most $\frac1{n-1}$, which occurs when the points are evenly spaced. To calculate $EM$, we first calculate $P(M>m)$, for $0\le m \le 1/(n-1)$.

$P(M>m)$ is, by symmetry, equal to $n!$ times $P(M>m\text{ and } U_1<U_2<\dots<U_n)$, and the probability of the latter is the volume of the below subset $S$ of the unit hypercube $Q=[0,1]^n$: $$ S = \{(x_1,\dots,x_n)\in Q:x_i+m\le x_{i+1}\text{ for }i=1,\dots,n-1\} $$ Now, consider the transformation $f:S\to [0,1]^n$, given by $$ (x_1,\dots,x_n)\mapsto (x_1,x_2-m,x_3-2m,\dots,x_n-(n-1)m) $$ A little thought shows that $f$ is a volume-preserving bijection from $S$ to the region $S'$ of the smaller hypercube $Q'=[0,1-(n-1)m]^n$ given by $$ S'=\{(y_1,\dots,y_n)\in Q':y_i\le y_{i+1}\text{ for }i=2,\dots,n-1\} $$ The volume of $S'$ is, by symmetry of permuting the $y_i$, equal to $1/n!$ times the volume of $Q'$, which is just $(1-(n-1)m)^n$. Putting this all together, $$ P(M>m)=n!\cdot \text{Vol}(S)=n!\cdot \frac1{n!}(1-(n-1)m)^n=(1-(n-1)m)^n $$ and therefore $$ \begin{align} EM =\int_0^{1/(n-1)}P(M>m)\,dm &=\int_0^{1/(n-1)}(1-(n-1)m)^n\,dm\\ &=\frac{1}{n-1}\frac{(1-(n-1)m)^{n+1}}{n+1}\Big|_{0}^{1/(n-1)}\\ &=\boxed{\frac{1}{n^2-1}} \end{align} $$ To get the answer for the interval $[0,a]$, simply multiply this result by $a$.