Determinants of random binary matrices

I ran across The determinant game on Alex Gittens’ blog ChapterZero where there was a comment pointing out that the problem is similar to problem A4 on Putnam competition.

The version of the determinant game in the Putnam problem is to start with an empty square matrix. Player 1 puts down a 1 somewhere, then player 0 puts down a 0 in an empty space and the two players then alternate turns. When there are no empty entries left, the determinant of the matrix is computed and if it is 0, then player 0 wins otherwise player 1 wins.

Since an optimal game of putting down and can be thought of as a particular subspace of a randomly constructed matrix composed of exactly entries equal to and the rest equal to , this got me thinking about random matrices.

Problem: Let be an matrix with exactly entries equal to and the other entries . If the location of the entries is uniformly chosen in the matrix , what is the probability that the determinant .

You can think of the problem as two dumb machines playing the above Determinant game, what is the probability that one will win?

I don’t have the solution, and I suspect it is not a trivial problem. Please let me know your thoughts, comments, solutions etc. Here are some easy calculations

for and for .
In the first case there aren’t enough s (at least one in each column) and in the second case again by pigeonhole principle there will be at least two columns composed of all 1s and therefore won’t be linearly independent.

Once the first 1 is put down anywhere, that column and row is scratched out and the next 1 must be put in the reduced matrix for linear independence.

This is similar to the previous calculation except with zeros. Notice that the product goes only up to .

Now only the interesting cases remain