Eigenvalue distribution of a band matrix

Question

Let $\mathbf M_i$ be rectangular matrices of dimensions $N_{i-1}\times N_i$. We assume that their entries are random, with zero mean and variance $\sigma_i^2$.

For some positive integer $k$, I define the matrix $\mathbf{A}_k$:

$$\mathbf{A}_k = \left(\begin{array}{cccccc} 0 & \mathbf{M}_1 & 0 & \cdots & 0 & 0\\ \mathbf{M}_1^{\top} & 0 & \mathbf{M}_2 & \cdots & 0 & 0\\ 0 & \mathbf{M}_2^{\top} & 0 & \ddots & 0 & 0\\ \vdots & \vdots & \ddots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \cdots & 0 & \mathbf{M}_n\\ 0 & 0 & 0 & \cdots & \mathbf{M}_k^{\top} & 0 \end{array}\right)$$

Notice that $\mathbf{A}$ is symmetric. But unfortunately it is not from a rotationally invariant random ensemble, which has complicated things for me.

I have two questions.

1. What is the eigenvalue density of $\mathbf{A}_k$?
2. What happens to the eigenvalue density when I set $N_i=c_iN$, with positive constants $c_i$ and we take the limit $N\rightarrow\infty$? I presume one must perform an appropriate re-scaling here to get a meaningful result.

Even if the answer to 1. is not tractable, maybe the limiting form 2. can still be obtained.

Update: Say I have access to the singular value decomposition of $\mathbf{M}_i$. Can we say anything about the eigenvalues/eigenvectors of $\mathbf{A}_k$?

Answer 1

1. A closed form expression for finite $N$ is most certainly not available.

2. A large-$N$ calculation for Gaussian distributed matrix elements has been reported in Density of eigenvalues of random band matrices. If the band width $\delta N$ increases with $N$ as $N^\beta$ for some $\beta>0$, the eigenvalue density tends in the limit $N\rightarrow\infty$ to the usual semicircle that one would obtain for the full Gaussian matrix (so for $\delta N=N$).