Consider an $n\times n$ random binary matrix $M$ with i.i.d. entries $m_{ij} \sim {\rm Bernoulli}(p)$, where $p = n^{-\beta}$ with $\beta \in (1,2)$. I am interested in the behavior of the singular value decomposition of $$M = \sum_{i=1}^{rank(M)} \sigma_i u_i v_i',$$ where $\sigma_i$ are ranked in decreasing order.

Some intuitive observation (which might NOT be all true!):

1) By subtracting the mean we can write $M = p {\bf 1} {\bf 1}' + A$, where $A$ has indepedent entries with zero mean and variance $p(1-p)$. Therefore I expect the leading singular vector is approximately parallel to the all-one vector ${\bf 1}$, and the largest singular value is $\sigma_1 \approx p n$. If the SVD of $A$ behaves similarly to that of the usual iid matrices, it is probably true that the second largest singular value of $M$ (i.e., the largest singular value of $A$) is approximately $\sigma_2 \approx \sqrt{p n}$.

2) ~~ $rank(M)$ is pretty small: since $\mathbb{P}(\text{the first}~ m \text{ rows are all zero}) = (1-p)^{n m}\geq 1-pmn$. Therefore $rank(M) \leq n^{\beta-1}$ with high probability. ~~ This is wrong... this only says that $rank(M) \leq n-n^{\beta-1}$.

Are there any rigorous results about the SVD of this matrix ensemble? Is it true that except for $u_1,v_1$ which are approximately $\frac{1}{\sqrt{n}} {\bf 1}$, the remaining singular vectors are independently and uniformly distributed over $S^{n-1}$?