Probability Theory and Related Fields

, Volume 167, Issue 1–2, pp 143–200 | Cite as

On the singularity of adjacency matrices for random regular digraphs

Article
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Abstract

We prove that the (non-symmetric) adjacency matrix of a uniform random d-regular directed graph on n vertices is asymptotically almost surely invertible, assuming \(\min (d,n-d)\ge C\log ^2n\) for a sufficiently large constant \(C>0\). The proof makes use of a coupling of random regular digraphs formed by “shuffling” the neighborhood of a pair of vertices, as well as concentration results for the distribution of edges, proved in Cook (Random Struct Algorithms. arXiv:1410.5595, 2014). We also apply our general approach to prove asymptotically almost surely invertibility of Hadamard products \(\varSigma {{\mathrm{\circ }}}\varXi \), where \(\varXi \) is a matrix of iid uniform \(\pm 1\) signs, and \(\varSigma \) is a 0/1 matrix whose associated digraph satisfies certain “expansion” properties.

Keywords

Random matrices Random regular digraphs Singularity probability Discrepancy property 

Mathematics Subject Classification

15B52 
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Notes

Acknowledgments

The author thanks Terence Tao for invaluable discussions on this problem and on random matrix theory in general, as well as for helpful feedback on preliminary versions of the manuscript. Thanks also go to Ioana Dumitriu and Jamal Najim for the suggestion to consider signed rrd matrices, in large part because the proof of Theorem 1.3 inspired arguments to improve the main theorem, allowing the degree d to lower from \(n^{1/2+\varepsilon }\) to \(C\log ^2n\). Finally, the author is grateful to the anonymous referees for their careful reading and numerous corrections and suggestions to improve the manuscript.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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