Expected minimum distance for $n$ points on a circle

Asked Jan 25 '20 at 09:43

Active Jan 25 '20 at 10:39

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Consider $n$ points chosen uniformly and independently on a circle of circumference $1$. What is the expected minimum (arc) distance between any pair of points? Clearly this is at most $1/n$.

This problem comes under the heading order statistics. We could therefore answer the question exactly if we were looking for the minimum of $n$ samples from $[0,1]$ but I don't know how to answer my question above.

edited Jan 25 '20 at 10:39

asked Jan 25 '20 at 09:43

1

what have you tried? Where you get stuck? – Masacroso Jan 25 '20 at 09:44
Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Jan 25 '20 at 09:45
Look at https://math.stackexchange.com/a/3511155/413376 – NCh Jan 25 '20 at 09:53
@NCh Is it simply $1/n^2$? – Jan 25 '20 at 10:02
Yes, sure. You can also look at this more general answer https://math.stackexchange.com/a/3260895/ – NCh Jan 25 '20 at 10:05
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@NCh If you increase the dimension so you have points on a sphere, do you get the same $1/n^2$ dependence? – Jan 25 '20 at 10:08
I do not know. How does it connected to a question? – NCh Jan 25 '20 at 10:17
@fomin Do you want the straight line distance or the arc length distance? (the answers will be different) – almagest Jan 25 '20 at 10:19
@almagest Arc length distance. – Jan 25 '20 at 10:20
@fomin Then please amend the question accordingly. At moment many readers will take it to be straight line distance. – almagest Jan 25 '20 at 10:22
@NCh It's the natural extension to 3d I believe. I may ask a separate question in this case. – Jan 25 '20 at 10:42
about the uniform distribution in the circle take a look here – Masacroso Jan 25 '20 at 11:36

Expected minimum distance for $n$ points on a circle

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