Let denote a random symmetric -matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take value and with probability ). Improving the earlier result by Costello, Tao, and Vu , we show that is nonsingular with probability for any positive constant . The proof uses an inverse Littlewood–Offord result for quadratic forms, which is of interest of its own.
Hoi H. Nguyen. "Inverse Littlewood–Offord problems and the singularity of random symmetric matrices." Duke Math. J. 161 (4) 545 - 586, 15 March 2012. https://doi.org/10.1215/00127094-1548344