(This is a follow-up to [The spectral radius of a binary matrix](https://mathoverflow.net/questions/267958/the-spectral-radius-of-a-binary-matrix))


Let $\mathcal B\_n$ denote the set of $n\times n$ matrices with entries in $\{0,1\}$. 


**QUESTION.** Is there a $\delta\in\bigl(0,\frac12\bigr), \alpha\in(0,1)$ and $b>0$ such that for any $n\ge1$ and any $M\in\mathcal B\_n$ with the number of 0s less than $\delta n^2$ we have $\rho(M)\ge bn^\alpha$?


If so, what's the best known value of $\delta$?


The **motivation** for this question comes from symbolic dynamics (hence the tags). Namely, is it true that for any topological Markov chain given by a matrix with "sufficiently many" 1s its topological entropy is positive? 



The spectral radius of a binary matrix - polynomial growth?
A paper that seems to directly address your question is the 1987 paper of Brualdi and Solheid, \*On the minimal spectral radius of matrices of zeros and ones". That paper shows that if the number of 1's is $(\frac12+\delta)n^2$, then the spectral radius is essentially $2\delta n$. The paper also allows you to answer the question of how many 1's you do need if you want sublinear spectral radius. The extremal examples are the 0--1 matrices that are all 1's below the diagonal and have square blocks of 1's centred around the diagonal. That is, you have a number of completely connected components, all of the same size (with one remainder component), and each component had edges only to later components. 


