Probability Theory and Related Fields

, Volume 173, Issue 3–4, pp 1301–1347 | Cite as

The smallest singular value of a shifted d-regular random square matrix

  • Alexander E. LitvakEmail author
  • Anna Lytova
  • Konstantin Tikhomirov
  • Nicole Tomczak-Jaegermann
  • Pierre Youssef


We derive a lower bound on the smallest singular value of a random d-regular matrix, that is, the adjacency matrix of a random d-regular directed graph. Specifically, let \(C_1<d< c n/\log ^2 n\) and let \(\mathcal {M}_{n,d}\) be the set of all \(n\times n\) square matrices with 0 / 1 entries, such that each row and each column of every matrix in \(\mathcal {M}_{n,d}\) has exactly d ones. Let M be a random matrix uniformly distributed on \(\mathcal {M}_{n,d}\). Then the smallest singular value \(s_{n} (M)\) of M is greater than \(n^{-6}\) with probability at least \(1-C_2\log ^2 d/\sqrt{d}\), where c, \(C_1\), and \(C_2\) are absolute positive constants independent of any other parameters. Analogous estimates are obtained for matrices of the form \(M-z\,\mathrm{Id}\), where \(\mathrm{Id}\) is the identity matrix and z is a fixed complex number.


Adjacency matrices Anti-concentration Condition number Invertibility Littlewood–Offord theory Random graphs Random matrices Regular graphs Singular probability Singularity Sparse matrices Smallest singular value 

Mathematics Subject Classification

Primary: 60B20 15B52 46B06 05C80 Secondary: 46B09 60C05 



We are grateful to an anonymous referee for careful reading the first draft of the manuscript and many valuable suggestions, which helped us to improve presentation. The second and the third named authors would like to thank University of Alberta for excellent working conditions in January–August 2016, when a significant part of this work was done.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Alexander E. Litvak
    • 1
    Email author
  • Anna Lytova
    • 2
  • Konstantin Tikhomirov
    • 3
  • Nicole Tomczak-Jaegermann
    • 1
  • Pierre Youssef
    • 4
  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Faculty of Mathematics, Physics, and Computer ScienceUniversity of OpoleOpolePoland
  3. 3.Department of MathematicsPrinceton UniversityPrincetonUSA
  4. 4.Laboratoire de Probabilités, Statistique et ModélisationUniversité Paris DiderotParisFrance

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