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

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Alexander E. Litvak & Nicole Tomczak-Jaegermann

  2. Faculty of Mathematics, Physics, and Computer Science, University of Opole, 48, Oleska Str., 45-052, Opole, Poland

    Anna Lytova

  3. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ, 08544, USA

    Konstantin Tikhomirov

  4. Laboratoire de Probabilités, Statistique et Modélisation, Université Paris Diderot, 75013, Paris, France

    Pierre Youssef

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  1. Alexander E. Litvak
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  4. Nicole Tomczak-Jaegermann
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Correspondence to Alexander E. Litvak.

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Litvak, A.E., Lytova, A., Tikhomirov, K. et al. The smallest singular value of a shifted d-regular random square matrix. Probab. Theory Relat. Fields 173, 1301–1347 (2019). https://doi.org/10.1007/s00440-018-0852-y

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