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):

The infinite norm of $\hat{\bf H}$ is obviously 1, and I know spectral radius is no larger than any natural norm. My question is how do I prove the spectral radius of this matrix is smaller than 1?

I did a simple numerical experiment, and found the claim should hold. If $p \to \infty$ and $\hat{N} \to \infty$, then the spectral radius should approach to 1.

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