The least singular value of a random symmetric matrix

Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe

Research output: Working paper/PreprintPreprint

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Abstract

Let A be a n×n symmetric matrix with (Ai,j)i≤j, independent and identically distributed according to a subgaussian distribution. We show that
ℙ(σmin(A)≤ε/√n)≤Cε+e−cn,
where σmin(A) denotes the least singular value of A and the constants C,c>0 depend only on the distribution of the entries of A. This result confirms a folklore conjecture on the lower-tail asymptotics of the least singular value of random symmetric matrices and is best possible up to the dependence of the constants on the distribution of Ai,j. Along the way, we prove that the probability A has a repeated eigenvalue is e−Ω(n), thus confirming a conjecture of Nguyen, Tao and Vu.
Original languageEnglish
PublisherarXiv
DOIs
Publication statusPublished - 11 Mar 2022

Bibliographical note

44 pages + 30 page supplement. The supplement builds on our previous work arXiv:2105.11384 and provides the proof of a technical quasi-randomness statement

Keywords

  • math.PR
  • math.CO

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