Maximum singular value of a random $\pm 1$ matrix 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.
 A: http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf
Theorem 5.39 (page 23) gives a non-asymptotic upper bound on the largest singular value
A: For what it's worth, it's relatively easy to simulate these in Mathematica, via
Manipulate[
 Histogram[
  N[SingularValueList[ 2 RandomInteger[{0, 1}, {m, n}] - 1, 1][[
      1]]] & /@ Range[10^power]], {m, 1, 10, 1}, {n, 1, 10, 
  1}, {power, 1, 6, 1}]

I've been playing around with it, and this 
 
is a histogram for 1,000,000 tries with $m=9$, $n=5$ for the five different singular values (each in a different colour). I'm intrigued by the peaks - were you expecting that? There is also a significant portion of matrices with one zero singular value, but I am unsure whether it is due to numerical artifacts.
A: 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.
