We have $N$ points randomly and uniformly distributed on a ring of length 1. Let $d_i$ be the distance between point $i$ and its first neighbor. We want to know the expected value of the smallest $d_i$, $E([d_i]_{min})$, being specially interested in the case of a quite small value of $N$ ($N<10$).

From numerical simulation it looks like $E([d_i]_{min}^{(N)})=N^{-2}$.

I faced problems in getting rid of correlations. As a first step, I tried to consider neighbors on the right side only, looking for the minimum of $N$ quantities $d_i^{(R)}$ that I suppose to be uncorrelated, but this is not a good approximation for small $N$.

I would be grateful for any help. Thank you

Luce