Consider a 0-1 integer $n \times n$ matrix with coefficients chosen uniformly over $\{0,1\}$. The probability that it is singular is exponentially small, and so we expect that it has a well-defined condition number.
What is the expected condition number — or better yet, what probabilistic upper bounds are there on the condition number (for constant or better probability) — for such a matrix? Equivalently: given that we might naively expect $n/2$ to be close to being an eigenvalue (with the all-1s vector being the eigenvector), what sort of behaviour do we expect of the smallest singular value of a random 0-1 matrix?