Let M_n be a uniformly-chosen random symmetric n x n matrix with entries in {-1,1}. What is the probability for det(M_n)=0? A wellknown conjecture states that the probability of this event is asymptotically equal to the probability that two of the rows or columns of M_n are equal (up to a factor of +-1) and hence is equal to \Theta(n^2 2^{-n}).
We developed an inverse Littlewood-Offord theorem in Z^n_p that applies under very mild conditions and made progress towards this conjecture, showing that the probability is bounded by exp(-c\sqrt{n}).
Joint work with Marcelo Campos, Robert Morris and Natasha Morrison.