Skip to main content
SpringerLink
Log in

Self-Avoiding Walks and Connective Constants

  • Conference paper
  • First Online:
Sojourns in Probability Theory and Statistical Physics - III

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 300))

Abstract

The connective constant \(\mu (G)\) of a quasi-transitive graph G is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on G from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph G.

  • We present upper and lower bounds for \(\mu \) in terms of the vertex-degree and girth of a transitive graph.

  • We discuss the question of whether \(\mu \ge \phi \) for transitive cubic graphs (where \(\phi \) denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles).

  • We present strict inequalities for the connective constants \(\mu (G)\) of transitive graphs G, as G varies.

  • As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator.

  • We describe so-called graph height functions within an account of ‘bridges’ for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function.

  • A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions.

  • Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not.

  • The review closes with a brief account of the ‘speed’ of SAW.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aizenman, M., Grimmett, G.R.: Strict monotonicity of critical points in percolation and ferromagnetic models. J. Stat. Phys. 63, 817–835 (1991)

    Article  MathSciNet  Google Scholar 

  2. Alm, S.E., Janson, S.: Random self-avoiding walks on one-dimensional lattices. Commun. Stat. Stoch. Models 6, 169–212 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babai, L.: Vertex-transitive graphs and vertex-transitive maps. J. Graph Theory 15, 587–627 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babai, L.: Automorphism groups, isomorphism, reconstruction. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. II, pp. 1447–1540. Elsevier, Amsterdam (1995)

    Google Scholar 

  5. Balister, P., Bollobás, B., Riordan, O.: Essential enhancements revisited (2014). https://arxiv.org/abs/1402.0834

  6. Bauerschmidt, R., Brydges, D.C., Slade, G.: Renormalisation group analysis of 4D spin models and self-avoiding walk. In: Proceedings ICMP, Santiago de Chile (2015). https://arxiv.org/abs/1602.04048

  7. Bauerschmidt, R., Duminil-Copin, H., Goodman, J., Slade, G.: Lectures on self-avoiding walks. In: Ellwood, D., Newman, C.M., Sidoravicius, V., Werner, W. (eds.) Probability and Statistical Physics in Two and More Dimensions, Clay Mathematics Institute Proceedings, vol. 15, pp. 395–476. CMI/AMS publication (2012)

    Google Scholar 

  8. Benjamini, I.: Self avoiding walk on the seven regular triangulations (2016). https://arxiv.org/abs/1612.04169

  9. Benjamini, I., Nachmias, A., Peres, Y.: Is the critical percolation probability local? Probab. Theory Relat. Fields 149, 261–269 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Benjamini, I., Schramm, O.: Percolation beyond \(\mathbb{Z}^{d}\), many questions and a few answers. Electron. Commun. Probab. 1, 71–82 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Benjamini, I., Schramm, O.: Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001). Article 23

    Google Scholar 

  12. Bezuidenhout, C., Grimmett, G.R., Kesten, H.: Strict inequality for critical values of Potts models and random-cluster processes. Commun. Math. Phys. 158, 1–16 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Boutillier, C., de Tilière, B.: The critical Z-invariant Ising model via dimers: locality property. Commun. Math. Phys. 301, 473–516 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Day, M.M.: Amenable semigroups. Illinois J. Math. 1, 509–544 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  15. Diestel, R., Leader, I.: A conjecture concerning a limit of non-Cayley graphs. J. Alg. Comb. 14, 17–25 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Am. Math. Soc. 284, 787–794 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duminil-Copin, H., Glazman, A., Hammond, A., Manolescu, I.: On the probability that self-avoiding walk ends at a given point. Ann. Probab. 44, 955–983 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Duminil-Copin, H., Hammond, A.: Self-avoiding walk is sub-ballistic. Commun. Math. Phys. 324, 401–423 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Duminil-Copin, H., Smirnov, S.: The connective constant of the honeycomb lattice equals \(\sqrt{2+\sqrt{2}}\). Ann. Math. 175, 1653–1665 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dvořák, Z., Mohar, B.: Spectral radius of finite and infinite planar graphs and of graphs of bounded genus. J. Comb. Theory Ser. B 100, 729–739 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fisher, M.E.: On the dimer solution of planar Ising models. J. Math. Phys. 7, 1776–1781 (1966)

    Article  Google Scholar 

  22. Flory, P.: Principles of Polymer Chemistry. Cornell University Press, Ithaca (1953)

    Google Scholar 

  23. Frauenkron, H., Causo, M.S., Grassberger, P.: Two-dimensional self-avoiding walks on a cylinder. Phys. Rev. E 59, R16–R19 (1999)

    Article  Google Scholar 

  24. Gilch, L.A., Müller, S.: Counting self-avoiding walks on free products of graphs. Discrete Math. 340, 325–332 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Grimmett, G.R.: Potts models and random-cluster processes with many-body interactions. J. Stat. Phys. 75, 67–121 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Grimmett, G.R.: Percolation, 2nd edn. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  27. Grimmett, G.R.: The Random-Cluster Model. Springer, Berlin (2006). http://www.statslab.cam.ac.uk/~grg/books/rcm.html

  28. Grimmett, G.R.: Three theorems in discrete random geometry. Probab. Surv. 8, 403–441 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Grimmett, G.R., Holroyd, A.E., Peres, Y.: Extendable self-avoiding walks. Ann. Inst. Henri Poincaré D 1, 61–75 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Grimmett, G.R., Li, Z.: Counting self-avoiding walks (2013). https://arxiv.org/abs/1304.7216

  31. Grimmett, G.R., Li, Z.: Self-avoiding walks and the Fisher transformation. Electron. J. Comb. 20, 14 p. (2013). Paper 47

    Google Scholar 

  32. Grimmett, G.R., Li, Z.: Strict inequalities for connective constants of transitive graphs. SIAM J. Discrete Math. 28, 1306–1333 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Grimmett, G.R., Li, Z.: Bounds on the connective constants of regular graphs. Combinatorica 35, 279–294 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Grimmett, G.R., Li, Z.: Cubic graphs and the golden mean. Discret. Math. (2019)

    Google Scholar 

  35. Grimmett, G.R., Li, Z.: Connective constants and height functions for Cayley graphs. Trans. Am. Math. Soc. 369, 5961–5980 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Grimmett, G.R., Li, Z.: Self-avoiding walks and amenability. Electron. J. Comb. 24, 24 p. (2017). Paper P4.38

    Google Scholar 

  37. Grimmett, G.R., Li, Z.: Locality of connective constants. Discret. Math. 341, 3483–3497 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Grimmett, G.R., Li, Z.: Weighted self-avoiding walks. J. Algebraic Comb. (2019)

    Google Scholar 

  39. Guttmann, A.J., Parviainen, R., Rechnitzer, A.: Self-avoiding walks and trails on the \(3.12^2\) lattice. J. Phys. A: Math. Gen. 38, 543–554 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Gwynne, E., Miller, J.: Convergence of the self-avoiding walk on random quadrangulations to SLE\(_{8/3}\) on \(\sqrt{8/3}\)-Liouville quantum gravity (2016). https://arxiv.org/abs/1608.00956

  41. Hammersley, J.M.: Percolation processes II. The connective constant. Proc. Camb. Phil. Soc. 53, 642–645 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  42. Hammersley, J.M., Morton, W.: Poor man’s Monte Carlo. J. R. Stat. Soc. B 16, 23–38 (1954)

    MathSciNet  MATH  Google Scholar 

  43. Hammersley, J.M., Welsh, D.J.A.: Further results on the rate of convergence to the connective constant of the hypercubical lattice. Q. J. Math. 13, 108–110 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  44. Jensen, I.: A parallel algorithm for the enumeration of self-avoiding polygons on the square lattice. J. Phys. A: Math. Gen. 36, 5731–5745 (2003)

    Article  MathSciNet  Google Scholar 

  45. Jensen, I.: Improved lower bounds on the connective constants for two-dimensional self-avoiding walks. J. Phys. A: Math. Gen. 37, 11521–11529 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. Jensen, I., Guttman, A.J.: Self-avoiding walks, neighbour-avoiding walks and trails on semiregular lattices. J. Phys. A: Math. Gen. 31, 8137–8145 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kesten, H.: Full Banach mean values on countable groups. Math. Scand. 7, 146–156 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kesten, H.: Symmetric random walks on groups. Trans. Am. Math. Soc. 92, 336–354 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kesten, H.: On the number of self-avoiding walks. J. Math. Phys. 4, 960–969 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  50. Lacoin, H.: Existence of a non-averaging regime for the self-avoiding walk on a high-dimensional infinite percolation cluster. J. Stat. Phys. 154, 1461–1482 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. Lacoin, H.: Non-coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster. Probab. Theory Relat. Fields 159, 777–808 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lawler, G., Schramm, O., Werner, W.: On the scaling limit of planar self-avoiding walk. In: Proceedings of Symposium Pure Mathematics, vol. 72, pp. 339–364 (2004)

    Google Scholar 

  53. Li, Z.: Local statistics of realizable vertex models. Commun. Math. Phys. 304, 723–763 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  54. Li, Z.: Critical temperature of periodic Ising models. Commun. Math. Phys. 315, 337–381 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  55. Li, Z.: Positive speed self-avoiding walks on graphs with more than one end (2016). https://arxiv.org/abs/1612.02464

  56. Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press, New York (2016). http://mypage.iu.edu/~rdlyons/

  57. Madras, N., Slade, G.: Self-Avoiding Walks. Birkhäuser, Boston (1993)

    MATH  Google Scholar 

  58. Madras, N., Wu, C.: Self-avoiding walks on hyperbolic graphs. Comb. Probab. Comput. 14, 523–548 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  59. Martineau, S.: The set of connective constants of Cayley graphs contains a Cantor space. Electron. Comm. Probab. 22 (2017). Paper No. 12

    Google Scholar 

  60. Martineau, S., Tassion, V.: Locality of percolation for Abelian Cayley graphs. Ann. Probab. 45, 1247–1277 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  61. Mohar, B.: Isoperimetric inequalities, growth, and spectrum of graphs. Lin. Alg. Appl. 103, 119–131 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  62. Mohar, B.: Some relations between analytic and geometric properties of infinite graphs. Discret. Math. 95, 193–219 (1991). Directions in infinite graph theory and combinatorics, Cambridge (1989)

    Article  MathSciNet  MATH  Google Scholar 

  63. Nachmias, A., Peres, Y.: Non-amenable Cayley graphs of high girth have \(p_c < p_u\) and mean-field exponents. Electron. Commun. Probab. 17, 1–8 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  64. Neumann, J.v.: Zur allgemeinen Theorie des Masses. Fund. Math. 13, 73–116 (1929)

    Google Scholar 

  65. Nienhuis, B.: Exact critical points and critical exponents of \(\rm O\)(\(n\)) models in two dimensions. Phys. Rev. Lett. 49, 1062–1065 (1982)

    Article  MathSciNet  Google Scholar 

  66. Orr, W.J.C.: Statistical treatment of polymer solutions at infinite dilution. Trans. Faraday Soc. 43, 12–27 (1947)

    Article  Google Scholar 

  67. Pak, I., Smirnova-Nagnibeda, T.: On non-uniqueness of percolation on non-amenable Cayley graphs. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 33, 495–500 (2000)

    Article  MATH  Google Scholar 

  68. Pönitz, A., Tittmann, P.: Improved upper bounds for self-avoiding walks on \(\mathbb{Z}^d\). Electron. J. Comb. 7 (2000). Paper R21

    Google Scholar 

  69. Renault, D.: The vertex-transitive TLF-planar graphs. Discret. Math. 309, 2815–2833 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  70. Schonmann, R.H.: Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Commun. Math. Phys. 219, 271–322 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  71. Thom, A.: A remark about the spectral radius. Int. Math. Res. Notices 10, 2856–2864 (2015)

    MathSciNet  MATH  Google Scholar 

  72. Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge University Press, Cambridge (2000)

    Book  MATH  Google Scholar 

  73. Zeilberger, D.: Self-avoiding walks, the language of science, and Fibonacci numbers. J. Stat. Plann. Inference 54, 135–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Engineering and Physical Sciences Research Council under grant EP/I03372X/1. GRG acknowledges valuable conversations with Alexander Holroyd concerning Questions 7 and 10, and the hospitality of UC Berkeley during the completion of the work. ZL acknowledges support from the Simons Foundation under grant #351813, and the National Science Foundation under grant DMS-1608896.

Author information

Authors and Affiliations

  1. Statistical Laboratory, Centre for Mathematical Sciences, Cambridge University, Wilberforce Road, Cambridge, CB3 0WB, UK

    Geoffrey R. Grimmett

  2. Department of Mathematics, University of Connecticut, 341 Mansfield Road U1009, Storrs, CT, 06269–1009, USA

    Zhongyang Li

Authors
  1. Geoffrey R. Grimmett
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhongyang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Geoffrey R. Grimmett .

Editor information

Editors and Affiliations

  1. NYU Shanghai, Shanghai, China

    Vladas Sidoravicius

Additional information

Dedicated to Chuck Newman in friendship on his 70th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Grimmett, G.R., Li, Z. (2019). Self-Avoiding Walks and Connective Constants. In: Sidoravicius, V. (eds) Sojourns in Probability Theory and Statistical Physics - III. Springer Proceedings in Mathematics & Statistics, vol 300. Springer, Singapore. https://doi.org/10.1007/978-981-15-0302-3_8

Download citation

Publish with us

Policies and ethics

Search

Navigation