Asymptotics for higher moments of random polyhedral cone angles

Let $m(d,k)$ denote the $k$-th moment of the random solid angle associated with the model, and let $d$ tend to infinity while $k$ is fixed. For every fixed $kin \{2,3,\ldots\}$, the constants $c_k$ are positive.

Higher-moment asymptotics. For every fixed $k\in \{2,3,\ldots\}$, we have

$$m(d,k) \sim \frac{c_k}{2^d\cdot d}$$

as $d\to\infty$, where $c_k>0$ is a constant. In particular, $c_2=1$ and $c_3=(12-\pi^2)/8$, whereas $m(d,1)=2^{-d}$.

This conjecture predicts the scale of all fixed higher moments in high dimension and reflects the contribution of configurations close to linear dependence. Its resolution is not indicated in the supplied text.