Let $m$ and $n$ be large positive integers (going to infinity), and let $W$ be a random matrix of size $n \times m$ with iid entries from $N(0,1/m)$. Let $x,y \in \mathbb R^m$ be deterministic vectors, and consider the angle $$ \theta_W(x,y) := \arccos\left(\frac{Wx}{\|Wx\|}\cdot \frac{Wy}{\|Wy\|}\right) $$ between the random $n$-dimensional Gaussian random vectors $Wx$ and $Wy$. Note that $\theta_W(x,y)$ is a random variable supported on $[0,\pi]$.
Question. What is the probability distribution $\theta_W(x,y)$ ?
Notes. I'm really only interested in the expectation of $\theta_W(x,y)$.
