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Your expression equals $$\frac{\sqrt{\pi } (n-1) \Gamma \left(\frac{n-1}{2}\right)}{4 \Gamma \left(\frac{n}{2}+1\right)},$$ which goes to zero as $n$ goes to infinity.

See this paper of R. E. Miles (he has plenty of related results for points on the sphere, etc, etc, mathscinet will tell you more). The results you want are in section 9 (p. 112, and thereabouts). (th …

The distribution of the maximal distance between a pair of random points on the circle is known - when you scale it by $n/\log n$ you get a Gumbel distribution with scale 1, location 1., see, e.g., …

This is equivalent to asking what the distribution of minimal interpoint distance is. This is addressed in a number of places, in particular, in this article of Tanagawa, Mochizuki, Tanaka, 1992.

This is really a follow-up on Suvrit's comment. There are plenty of formulas for products of theta functions, many of them found in this Iowa State report. (see particularly page 7). Whether any of th …

A very similar question is considered here. (here, the OP is asking about two segments in the unit circle). There are two possibilities: the four points form the vertices of a convex quadrilateral (in …

It seems that even the constant in front of the $\sqrt{n}$ is not known, but there are experimental results which seem to describe the distribution pretty well. In particular, it seems that the varian …