Asymptotic Metric Behavior of Random Cayley Graphs of Finite Abelian Groups


Using methods of Marklof and Strömbergsson we establish several limit laws for metric parameters of random Cayley graphs of finite abelian groups with respect to a randomly chosen set of generators of a fixed size. Doing so we settle a conjecture of Amir and Gurel- Gurevich.

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  1. [1]

    G. Amir and O. Gurel-Gurevich: The diameter of a random Cayley graph of Zq, Groups Complex. Cryptol.2 (2010), 59–65.

    MathSciNet  Article  Google Scholar 

  2. [2]

    L. Clozel, H. Oh and E. Ullmo: Hecke operators and equidistribution of Hecke points, Invent. Math.144 (2001), 327–351.

    MathSciNet  Article  Google Scholar 

  3. [3]

    M. Einsiedler, S. Mozes, N. Shah and U. Shapira: Equidistribution of primitive rational points on expanding horospheres, Compos. Math.152 (2016), 667–692.

    MathSciNet  Article  Google Scholar 

  4. [4]

    A. Eskin and H. Oh: Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems26 (2006), 163–167.

    MathSciNet  Article  Google Scholar 

  5. [5]

    D. Goldstein and A. Mayer: On the equidistribution of Hecke points, Forum Math.15 (2003), 165–189.

    MathSciNet  Article  Google Scholar 

  6. [6]

    J. Marklof: The asymptotic distribution of Frobenius numbers, Invent. Math.181 (2010), 179–207.

    MathSciNet  Article  Google Scholar 

  7. [7]

    J. Marklof and A. Strömbergsson: Diameters of random circulant graphs, Combinatorica33 (2013), 429–466.

    MathSciNet  Article  Google Scholar 

  8. [8]

    H. P. F. Swinnerton-Dyer: A brief guide to algebraic number theory, London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 50, 2001.

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Correspondence to Reut Zuck.

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Shapira, U., Zuck, R. Asymptotic Metric Behavior of Random Cayley Graphs of Finite Abelian Groups. Combinatorica 39, 1133–1148 (2019).

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Mathematics Subject Classification (2000)

  • 05C12
  • 05C80
  • 37A99