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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):

enter image description here

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

enter image description here

If we fix $\hat{N} = 10$ and let $p$ go from 5 to 100, we have

enter image description here

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  • $\begingroup$ What make you think that such a form exist ? It is very exceptional that such formulas exist... $\endgroup$ – Jean Marie Becker Dec 8 '18 at 6:11

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