# Distribution of $(\langle X_i,X_j\rangle)_{i,j=1}^n$ for $X_k\sim\operatorname{Unif}(S^d)$

For two independent random variables distributed uniformly on a $$d$$-sphere surface ($$X_1,X_2\sim\operatorname{Unif}(S^d)$$), it is obvious that $$\langle X_1,X_2\rangle\sim -\langle X_1,X_2\rangle$$ by symmetry of $$X_1$$ and linearity of the inner product. Can this notion be generalized to the joint distribution of the pairwise scalar combination of $$n$$ such variables by using symmetries in a clever way? Namely, does $$((1-\delta_{ij})\langle X_i,X_j\rangle)_{i,j=1}^n\sim (-(1-\delta_{ij})\langle X_i,X_j\rangle)_{i,j=1}^n$$ hold? By rotational invarince, one can derive the density of each such entry rather quick, but that does not seem helpful as the dependence/interaction is rather involved.

• What's the connection to random graphs? Is there one? – Misha Lavrov Jan 15 at 14:45
• I have not stated the detailed relation to random graphs since it's not crucial for the question asked (and i didn't want to include unnecessary information), although some may recognize a connection. However, i came across this issue when facing a random graph problem, where random adjacency matrices arise by applying indicators on the entries above (symmetry and a vanishing diagonal are already provided). – Fiff Jan 16 at 9:08
• That's interesting. If you place an edge $ij$ whenever $\langle X_i, X_j\rangle \ge \alpha$ for some threshold $\alpha$, then you get a random geometric graph in the $d$-sphere, for instance. – Misha Lavrov Jan 16 at 14:30
• That is exactly the context in which i am studying these type of matrices. I suspected that the total variation distance between the random geometric graph on the $d$-sphere und the unstructured Erdős–Rényi graph might be symmetric around single-edge probability $p=\frac{1}{2}$ and the above would have shown that. But apparently, this approach is just a dead end. – Fiff Jan 16 at 21:03

The thing you want just isn't true: if you have $$n=5$$ random variables distributed uniformly on $$S^1$$, for instance, then it's impossible for $$\langle X_i, X_j\rangle$$ to be negative for all pairs $$i\ne j$$. However, it's perfectly possible (and happens with probability better than $$\frac{1}{4^4}$$: the chances that all of $$X_1, X_2, \dots, X_5$$ land in the same quadrant) that $$\langle X_i, X_j\rangle$$ is positive for all pairs $$i \ne j$$.
(The same thing happens in any dimension if you take $$n$$ large enough as a function of $$d$$.)