I am studying a family of random matrices and observe a universal concentration behavior which I would like to understand.
Let $U$ be an $N \times N$ random unitary drawn from the Haar measure, and $Z = \text{diag}(1,1,1\cdots,-1,-1,-1\cdots)$ (equal ones and minus ones). Define the random variable $\eta(U) := U Z U^\dagger$.
Define the isometries $W = \begin{pmatrix} \mathbb{I}_{n,n} ~ 0_{n,N-n} \end{pmatrix}$ and $V = \begin{pmatrix} 0_{N-n,n} ~ \mathbb{I}_{N-n,N-n} \end{pmatrix}$ where the indices of the $0$ matrices and identity matrices denote their sizes. We have $W W^\dagger = \mathbb{I}_{n,n}$, $VV^\dagger =\mathbb{I}_{N-n,N-n}$, while $WW^\dagger, VV^\dagger$ are projectors onto the top left $n \times n$ block and bottom right $(N-n)\times (N-n)$ block respectively, so that $WW^\dagger +VV^\dagger = \mathbb{I}_{N,N}$.
I consider the $n \times n$ random matrix \begin{align} X_r:= W\eta V^T \left(V(r \eta + \mathbb{I}_{N,N})V^T\right)^{-1} V\eta W^T \end{align} for $r > 0$.
I numerically observe that, as long as $r \neq 1$ and fixing $n$, \begin{align} \lim_{N \to \infty} X_r \to \mathbb{I}_{n,n} \end{align} (I guess at least in probability), regardless of $r$. This is what I mean by "universal", in the title.
Here are some numerical plots illustrating this. I compute coeff = $\text{Tr}(X_r)$ (i.e., the Hilbert-Schmidt overlap of $X_s$ to $\mathbb{I}_{n,n}$, and I plot their distribution for various $N = 20, 40, 80$, fixing $n=2$:
These are very small $N$'s I am plotting, but visually, one can see there is a clear concentration of the distributions to a delta-function at $1$ as $N$ increases. Better evidence is seen, if I let $N$ be even bigger, say $N = 1000$ (not plotted), then I observe every instance of $X_r$ is really rather very close to the identity.
How do I understand this? Intuitively, it makes sense that there should be concentration, since $X_r$ is a small random matrix generated by "squeezing" a large random matrix $\eta$ ---- all the random numbers conspire to give a limiting behavior. But showing this more rigorously is eluding me.
From invariance of the Haar measure, I can see that $X_r$ is invariant under unitary conjugation, so if its expectation converges, it must be proportional to the identity, with the coefficient given by its trace. But how do I evaluate that? (Numerically it seems to be equal to $1$ for any $N$, in fact). And how do I understand the universal concentration of measure, as long as $r \neq 1$? I would be happy with any understanding, even asymptotics, no need for explicit computation.