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Let $P_1, P_2, ..., P_n$ be points randomly placed on a unit circle from a uniform distribution. Consider the product $D$ of all pairwise distances:

$D=\displaystyle \prod_{1\leq i < j \leq n} \overline{P_iP_j}$

I wonder...

1) What is the probability density function for $D$? What is the expected value of $D$?

2) When the $P_i$ are equally spaced, we know $D=n^{n/2}$. Is this an absolute maximum for $D$?

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To 2, the answer is yes. This is easy. Re-enumerate points according to their cyclic order. For every fixed $k$ ($1\le k<n$), the product $$ D_k:= \prod_i |P_i P_{i+k}| $$ (where the indices are taken modulo $n$), equals $$ D_k=\prod_i 2\sin (t_i/2) = 2^n\exp \left(\sum_i \log\sin(t_i/2) \right) $$ where $t_i$ is the (oriented) angle from $P_i$ to $P_{i+k}$. Since $t_i\in(0,2\pi)$, $\sum t_i=2\pi k$, and the function $t\mapsto \log\sin(t/2)$ is concave on $(0,2\pi)$, by Jensen's inequality the maximum is attained when all $t_i$ are equal, i.e., when the points are equally spaced. Since $D=\prod_k D_k^{1/2}$, the result follows.

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Not an answer to 2 (though I am guessing this is true and not very difficult). For 1, you are asking for the distribution of determinants of random Vandermonde matrices with entries i.i.d. on the unit circle. (write your product in terms of complex numbers). This very question has been studied by Oyvind Ryan, see these slides for some of the results and pointers to his actual papers.

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  • $\begingroup$ Your link is unfortunately dead. Are you aware of any bounds that don't arise in the "large-$N$" regime (i.e., expressions for squares and fourth powers of determinants of $N\times N$ Vandermonde matrices for finite $N$, not in the limit)? Also, Ryan seems to study the moments of these matrices in general, i.e. the traces, not determinants in his work. $\endgroup$ – Cornelius Brand Oct 30 '19 at 16:30

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