TITLE: Maximum singular value of a random $\pm 1$ matrix
QUESTION [11 upvotes]: Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in the literature that talks about properties of this kind of matrices?  
I have seen that there are some results for other kind of random matrices (for example matrices whose entries are i.i.d. Gaussian.) but not for this simple matrix of $\pm 1$.
I would be interested for example on the distribution of the $\sigma_{\max}(A)$, but not in an asymptotic regime, as $m$, $n$ are finite numbers and usually small in my case.
Thank you very much for any pointer or any thoughts.

REPLY [3 votes]: In relation to the first question, a recent breakthrough result of Konstantin Tikhomirov answers affirmatively an old conjecture attributed to Von Neumann, namely that for $M_n$, an $n\times n$ random matrix with independent $\pm 1$ entries
$$\mathbb{P}(M_{n}\ \text{is singular})=\left(\frac{1}{2}+o(1)\right)^n.$$
Here is the pre-print and a relevant entry in Gil Kalai's blog can be found here.