# Product of random diagonals on the unit circle

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 $D$?

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.
• 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. – Cornelius Brand Oct 30 '19 at 16:30