Singularity of random symmetric matrices revisited

Abstract

Let M n M_n be drawn uniformly from all ± 1 \pm 1 symmetric n × n n \times n matrices. We show that the probability that M n M_n is singular is at most exp ⁡ ( − c ( n log ⁡ n ) 1 / 2 ) \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 ) \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.