Electronic Communications in Probability

The set of connective constants of Cayley graphs contains a Cantor space

Sébastien Martineau

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Abstract

The connective constant of a transitive graph is the exponential growth rate of its number of self-avoiding walks. We prove that the set of connective constants of the so-called Cayley graphs contains a Cantor set. In particular, this set has the cardinality of the continuum.

Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 12, 4 pp.

Dates
Received: 26 August 2016
Accepted: 13 January 2017
First available in Project Euclid: 27 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1485507643

Digital Object Identifier
doi:10.1214/17-ECP43

Mathematical Reviews number (MathSciNet)
MR3607807

Zentralblatt MATH identifier
1357.82014

Subjects
Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Keywords
Cayley graph transitive graph connective constant uncountability

Rights
Creative Commons Attribution 4.0 International License.

Citation

Martineau, Sébastien. The set of connective constants of Cayley graphs contains a Cantor space. Electron. Commun. Probab. 22 (2017), paper no. 12, 4 pp. doi:10.1214/17-ECP43. https://projecteuclid.org/euclid.ecp/1485507643


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References

  • [1] László Babai,Vertex-transitive graphs and vertex-transitive maps, Journal of graph theory 15 (1991), no. 6, 587–627.
  • [2] Itai Benjamini and Oded Schramm,Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001), no. 23, 1–13.
  • [3] Pierre de la Harpe,Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000.
  • [4] Reinhard Diestel and Imre Leader,A conjecture concerning a limit of non-Cayley graphs, Journal of Algebraic Combinatorics 14 (2001), no. 1, 17–25.
  • [5] Geoffrey Grimmett and Zhongyang Li,Locality of connective constants, arXiv:1412.0150.
  • [6] Geoffrey Grimmett and Zhongyang Li,Strict inequalities for connective constants of transitive graphs, SIAM Journal on Discrete Mathematics 28 (2014), no. 3, 1306–1333.
  • [7] Gady Kozma, Personal communication.
  • [8] Imre Leader and Klas Markström,Uncountable families of vertex-transitive graphs of finite degree, Discrete mathematics 306 (2006), no. 7, 678–679.