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Let $k$ be any fixed constant, and suppose that your matrix contains no $k+1 \times k+1$ principal submatrix with at least $k$ nonzero entries. Then your matrix has a block decomposition with all blocks having size at most $k$ k+1$(If you start with a given diagonal nonzero entry and "grow" it by adding entries in the same row/column as an entry already added, you'll stop by the time you've reached$k-1$added entries, and each addition increases the size of your block by at most$1$). In this case we can upper bound the probably a submatrix of this size exists by $$\binom{n}{k+1} \binom{(k+1)^2}{k} p^{k} \leq C_k n^{k+1-k \beta}$$ Which goes to$0$for any$\beta$in your range if$k$sufficiently large. This means you'll usually see singular vectors with very small support, and that the$\sigma_2$should be much larger than$\sqrt{np}$(maybe equal to$\sigma_1$in most cases?) You might be able to get something more explicit by enumerating all the possible structures of blocks and the expected number of occurrences of each. 1 Also in the "a little too involved for a comment" class: A matrix that's this sparse is usually going to be a block diagonal matrix with very small blocks. Let$k$be any fixed constant, and suppose that your matrix contains no$k+1 \times k+1$principal submatrix with at least$k$nonzero entries. Then your matrix has a block decomposition with all blocks having size at most$k$(If you start with a given diagonal entry and "grow" it by adding entries in the same row/column as an entry already added, you'll stop by the time you've reached$k-1$added entries, and each addition increases the size of your block by at most$1$). In this case we can upper bound the probably a submatrix of this size exists by $$\binom{n}{k+1} \binom{(k+1)^2}{k} p^{k} \leq C_k n^{k+1-k \beta}$$ Which goes to$0$for any$\beta$in your range if$k$sufficiently large. This means you'll usually see singular vectors with very small support, and that the$\sigma_2$should be much larger than$\sqrt{np}$(maybe equal to$\sigma_1\$ in most cases?) You might be able to get something more explicit by enumerating all the possible structures of blocks and the expected number of occurrences of each.