A well-known conjecture states that a random symmetric matrix with entries in is singular with probability . We prove that the probability of this event is at most , improving the best-known bound of , which was obtained recently by Ferber and Jain. The main new ingredient is an inverse Littlewood–Offord theorem in that applies under very mild conditions, whose statement is inspired by the method of hypergraph containers.
"On the singularity of random symmetric matrices." Duke Math. J. 170 (5) 881 - 907, 1 April 2021. https://doi.org/10.1215/00127094-2020-0054