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Probability Theory and Related Fields

, Volume 156, Issue 3–4, pp 921–975 | Cite as

Functional limit theorems for random regular graphs

  • Ioana Dumitriu
  • Tobias JohnsonEmail author
  • Soumik Pal
  • Elliot Paquette
Article

Abstract

Consider \(d\) uniformly random permutation matrices on \(n\) labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree \(2d\) on \(n\) vertices. We consider limit theorems for various combinatorial and analytical properties of this graph (or the matrix) as \(n\) grows to infinity, either when \(d\) is kept fixed or grows slowly with \(n\). In a suitable weak convergence framework, we prove that the (finite but growing in length) sequences of the number of short cycles and of cyclically non-backtracking walks converge to distributional limits. We estimate the total variation distance from the limit using Stein’s method. As an application of these results we derive limits of linear functionals of the eigenvalues of the adjacency matrix. A key step in this latter derivation is an extension of the Kahn–Szemerédi argument for estimating the second largest eigenvalue for all values of \(d\) and \(n\).

Keywords

Random regular graphs Sparse random matrices  Poisson approximation Linear eigenvalue statistics Infinitely divisible distributions 

Mathematics Subject Classification (2000)

60B20 05C80 

Notes

Acknowledgments

The authors gratefully acknowledge several useful discussions with Gérard Ben Arous, Kim Dang, Joel Friedman, and Van Vu. In particular, they thank Gérard for sharing the article [4]. Joel and Van have generously pointed us toward the techniques in [21] that we use in Sect. 4. Additionally, we would like thank the referees and the associate editor for many insightful comments.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Ioana Dumitriu
    • 1
  • Tobias Johnson
    • 1
    Email author
  • Soumik Pal
    • 1
  • Elliot Paquette
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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