Singularity of random symmetric matrices revisited.

Let M_n be drawn uniformly from all \pm 1 symmetric n \times n matrices. We show that the probability that M_n is singular is at most \exp (-c(n\log n)^{1/2}), which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of \exp (-c n^{1/2}) on the singularity probability, our method is different and considerably simpler: we prove a "rough" inverse Littlewood-Offord theorem by a simple combinatorial iteration. [ABSTRACT FROM AUTHOR]