Comptes Rendus
Probabilités
Singularity of random symmetric matrices – simple proof
Asaf Ferber1 
1 Department of Mathematics, University of California, Irvine, USA.
Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 743-747.

In this paper we give a simple, short, and self-contained proof for a non-trivial upper bound on the probability that a random ±1 symmetric matrix is singular.

Reçu le :
Révisé le :
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DOI : 10.5802/crmath.215

Asaf Ferber 1

1 Department of Mathematics, University of California, Irvine, USA.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Asaf Ferber. Singularity of random symmetric matrices – simple proof. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 743-747. doi : 10.5802/crmath.215. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.215/

[1] Marcelo Campos; Letícia Mattos; Robert Morris; Natasha Morrison On the singularity of random symmetric matrices (2019) (https://arxiv.org/abs/1904.11478) | Zbl

[2] Kevin P. Costello; Terence Tao; Van H. Vu Random symmetric matrices are almost surely nonsingular, Duke Math. J., Volume 135 (2006) no. 2, pp. 395-413 | MR | Zbl

[3] Asaf Ferber; Vishesh Jain Singularity of random symmetric matrices—a combinatorial approach to improved bounds, Forum Math. Sigma, Volume 7 (2019), e22 | MR | Zbl

[4] Asaf Ferber; Vishesh Jain; Kyle Luh; Wojciech Samotij On the counting problem in inverse Littlewood–Offord theory (2019) (https://arxiv.org/abs/1904.10425) | Zbl

[5] Jiaoyang Huang Invertibility of adjacency matrices for random d-regular graphs (2018) (https://arxiv.org/abs/1807.06465)

[6] Hoi H Nguyen Inverse Littlewood–Offord problems and the singularity of random symmetric matrices, Duke Math. J., Volume 161 (2012) no. 4, pp. 545-586 | MR | Zbl

[7] Roman Vershynin Invertibility of symmetric random matrices, Random Struct. Algorithms, Volume 44 (2014) no. 2, pp. 135-182 | DOI | MR | Zbl

  • Asaf Ferber; Matthew Kwan; Ashwin Sah; Mehtaab Sawhney Singularity of the k-core of a random graph, Duke Mathematical Journal, Volume 172 (2023) no. 7 | DOI:10.1215/00127094-2022-0060
  • Asaf Ferber; Liam Hardiman; Michael Krivelevich On subgraphs with degrees of prescribed residues in the random graph, Random Structures Algorithms, Volume 63 (2023) no. 1, p. 192 | DOI:10.1002/rsa.21137
  • Asaf Ferber; Matthew Kwan; Lisa Sauermann Singularity of sparse random matrices: simple proofs, Combinatorics, Probability and Computing, Volume 31 (2022) no. 1, p. 21 | DOI:10.1017/s0963548321000146
  • Elad Aigner-Horev; Yury Person On sparse random combinatorial matrices, Discrete Mathematics, Volume 345 (2022) no. 11, p. 113017 | DOI:10.1016/j.disc.2022.113017
  • Matthew Kwan; Lisa Sauermann On the permanent of a random symmetric matrix, Selecta Mathematica, Volume 28 (2022) no. 1 | DOI:10.1007/s00029-021-00730-6

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