Average distance between two points on $U(n)$

Question

I'm interested in the following problem :

Compute the average distance between two points independently chosen at random on $U(n)$.

What i have done so far :

• Observe that : $$\begin{array}{lll} \int_{U(n)}\int_{U(n)}d(g_1,g_2)d\mu_1d\mu_2&=&\int\int d(g_1g_2^{-1},1)d\mu_1d\mu_2\\ &=&\int\int d(g,1) d\mu d\mu_2\\ &=&\int d(g,1) d\mu\\ \end{array}$$

where :

1. the second equality comes from the bi-invariance of the Haar-measure;
2. the last equality comes from the fact that $\mu_2({U(n)})=1$.

• Now, $g\mapsto d(g,1)$ is invariant by conjugation so Weyl's integration formula applies :

$$\int_{U(n)} d(g,1) d\mu = \int_{T} d(t,1) u(t) dt =\frac{1}{n!}\int_{[-\pi;\pi]^n} \sqrt{\theta_1^2+...+\theta_n^2}\prod_{i\neq j}\left|e^{i\theta_i}-e^{i\theta_j}\right|\frac{d\theta_1}{2\pi}...\frac{d\theta_n}{2\pi}$$

This is because, if $t=diag(e^{i\theta_1},...e^{i\theta_n})$ with $\theta_i\in[-\pi,\pi]$, then $d(t,1)=\sqrt{\theta_1^2+\ldots+\theta_n^2}$.

Indeed, $\gamma:[0,1]\to U(n)$, $\gamma(s)=diag(e^{is\theta_1},...e^{is\theta_n})$ is a minimal geodesic of length $||(\theta_1,...,\theta_n)||=\sqrt{\theta_1^2+...+\theta_n^2}$.

Questions:

1. Are my reasoning and computation so far correct ?
2. If they are, how can I compute that monstrosity ?! I mean, even in the simplest non trivial case where $n=2$ :

$$\int_{U(2)} d(g,1)d\mu = \frac{1}{2\pi^2}\int_{[-\pi;\pi]^2} \sqrt{x^2+y^2}\sin\left(\frac{x-y}{2}\right)^2 dx\,dy$$

Wolfram and back-of-the-envelope simulations with Sage suggest the result is $\approx 2,48$. Is this the best we can expect ?!

3. If exact computation is not manageable for general $n$, how can i compute at least an equivalent when $n\to \infty$ ?

4. I'd like to understand how much the unitary case deviates from abelian case of same rank. (Simulations on Sage suggest that the average distance between two points on the standard torus $T_n$ grows like $O(\sqrt{n})$.) How can I do that ?

5. Surely, these questions must have been investigated but i don't where to look for. What are the key-words/references for these questions ?

EDIT2: i think i can answer question 3. for the torus case !

Indeed, let $(X_i)_{i\in\mathbb{N}}$ be a sequence of i.i.d. variables with uniform distribution $\mathcal{U}([-\pi;\pi])$.

Then $\frac{X_1^2+\ldots+X_n^2}{n}\rightarrow^{a.s.} E[x_1^2]=\frac{\pi^2}{3}$ (law of large numbers).

Therefore, $$\int_{[-\pi,\pi]^n}\sqrt{x_1^2+...+x_n^2}\frac{dx_1}{2\pi}...\frac{dx_n}{2\pi} \sim \sqrt{n}\sqrt{\frac{\pi^2}{3}}=\frac{\pi}{\sqrt{3}}\sqrt{n}$$

If this is correct, this settles question 3. for the torus case.