1 April 2021 On the singularity of random symmetric matrices
Marcelo Campos, Letícia Mattos, Robert Morris, Natasha Morrison
Author Affiliations +
Duke Math. J. 170(5): 881-907 (1 April 2021). DOI: 10.1215/00127094-2020-0054


A well-known conjecture states that a random symmetric n×n matrix with entries in {1,1} is singular with probability Θ(n22n). We prove that the probability of this event is at most exp(Ω(n)), improving the best-known bound of exp(Ω(n14logn)), which was obtained recently by Ferber and Jain. The main new ingredient is an inverse Littlewood–Offord theorem in Zpn that applies under very mild conditions, whose statement is inspired by the method of hypergraph containers.


Download Citation

Marcelo Campos. Letícia Mattos. Robert Morris. Natasha Morrison. "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


Received: 27 April 2019; Revised: 9 April 2020; Published: 1 April 2021
First available in Project Euclid: 18 March 2021

Digital Object Identifier: 10.1215/00127094-2020-0054

Primary: 60B20
Secondary: 05D40 , 15B52

Keywords: containers , inverse Littlewood–Offord theorems , random symmetric matrices , singularity

Rights: Copyright © 2021 Duke University Press

Vol.170 • No. 5 • 1 April 2021
Back to Top