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Proceedings of the American Mathematical Society

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Singularity of random symmetric matrices revisited


Authors: Marcelo Campos, Matthew Jenssen, Marcus Michelen and Julian Sahasrabudhe
Journal: Proc. Amer. Math. Soc. 150 (2022), 3147-3159
MSC (2020): Primary 60B20
DOI: https://doi.org/10.1090/proc/15807
Published electronically: March 24, 2022
MathSciNet review: 4428895
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Abstract: 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.


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Additional Information

Marcelo Campos
Affiliation: Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil
MR Author ID: 1378562
Email: marcelo.campos@impa.br

Matthew Jenssen
Affiliation: School of Mathematics, University of Birmingham, Birmingham, United Kingdom
MR Author ID: 1015306
ORCID: 0000-0003-0026-8501
Email: m.jenssen@bham.ac.uk

Marcus Michelen
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607
MR Author ID: 1312016
Email: michelen.math@gmail.com

Julian Sahasrabudhe
Affiliation: Department of Pure Mathematics and Mathematics Statistics (DPMMS), University of Cambridge, Cambridge, United Kingdom
MR Author ID: 933725
Email: jdrs2@cam.ac.uk

Received by editor(s): January 17, 2021
Received by editor(s) in revised form: August 3, 2021
Published electronically: March 24, 2022
Additional Notes: The first author was partially supported by CNPq
The third author was partially supported by NSF grant DMS-2137623
Communicated by: Qi-Man Shao
Article copyright: © Copyright 2022 by the authors