Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (empty places are zero):
Asking for help to find the analytic form of the spectral radius (or eigenvalues) of this matrix. If analytic form is hard to calculate, would it be possible to at least determine the order of the spectral radius approaching 1? I did a simple numerical experiment as the following,
If we fix $p = 5$ and let $\hat{N}$ go from 5 to 100, we have
If we fix $\hat{N} = 10$ and let $p$ go from 5 to 100, we have
