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 of this matrix. If this is hard, then finding the order of the spectral radius approaching 1 is also good for us. 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