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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


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How is the question related to algebraic geometry? – Squark Dec 3 '12 at 12:41
up vote 3 down vote accepted

You can generate a point-set with the correct joint distribution of gap-sizes by running a rate 1 Poisson point process on the positive real numbers until you have $N$ events. Then wrap around to get a circle by identifying the time of the $N$th event with the origin.

By standard properties of the Poisson process, conditioning on the time of the $N$th point, the first $N-1$ points (forgetting the ordering) are distributed like $N-1$ independent uniform points in the interval from zero to the time of the $N$th point. Therefore, if we rescale the circle to perimeter 1, this is exactly what you want.

The smallest gap is the smallest of $N$ independent standard exponential variables, which has expectation $1/N$. And then there is a factor $1/N$ from the rescaling. So your guess from simulations is correct!

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PYKE, R. (1965). Spacings. J. R. Stat. Soc. Ser. B Stat. Methodol. 27 395–449.

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