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The Language of Self-Avoiding Walks

Abstract

Let X = (VX, EX) be an infinite, locally finite, connected graph without loops or multiple edges. We consider the edges to be oriented, and EX is equipped with an involution which inverts the orientation. Each oriented edge is labelled by an element of a finite alphabet Σ. The labelling is assumed to be deterministic: edges with the same initial (resp. terminal) vertex have distinct labels. Furthermore, it is assumed that the group of label-preserving automorphisms of X acts quasi-transitively. For any vertex o of X, consider the language of all words over Σ which can be read along self-avoiding walks starting at o. We characterize under which conditions on the graph structure this language is regular or context-free. This is the case if and only if the graph has more than one end, and the size of all ends is 1, or at most 2, respectively.

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References

  1. [1]

    S. E. Alm and S. Janson: Random self-avoiding walks on one-dimensional lattices, Comm. Statist. Stochastic Models6 (1990), 169–212.

    MathSciNet  Article  Google Scholar 

  2. [2]

    A. V. Anisimov: Group languages, Kibernetika4 (1971), 18–24.

    MathSciNet  Google Scholar 

  3. [3]

    R. Bauerschmidt, H. Duminil-Copin, J. Goodman and G. Slade: Lectures on self-avoiding walks, in: Probability and Statistical Physics in Two and more Dimensions, 395–467, Clay Math. Proc. 15, Amer. Math. Soc., Providence, RI, 2012.

    MathSciNet  MATH  Google Scholar 

  4. [4]

    T. Ceccherini-Silberstein, M. Coornaert, F. Fiorenzi, P. E. Schupp and N. W. M. Touikan: Multipass automata and group word problems, Theoret. Comput. Sci.600 (2015), 19–33.

    MathSciNet  Article  Google Scholar 

  5. [5]

    T. Ceccherini-Silberstein and W. Woess: Growth and ergodicity of context-free languages, Trans. Amer. Math. Soc.354 (2002), 4597–4625.

    MathSciNet  Article  Google Scholar 

  6. [6]

    T. Ceccherini-Silberstein and W. Woess: Context-free pairs of groups I: Context-free pairs and graphs, European J. Combin.33 (2012), 1449–1466.

    MathSciNet  Article  Google Scholar 

  7. [7]

    N. Chomsky and M.-P. Schützenberger: The algebraic theory of context-free languages, in: Computer Programming and Formal Systems26 (P. Braffort abd D. Hirschberg, eds.), North-Holland, Amsterdam, 1963, 118–161.

    MathSciNet  Article  Google Scholar 

  8. [8]

    W. Dicks and M. J. Dunwoody: Groups acting on graphs, Cambridge Studies in Advanced Mathematics17, Cambridge Univ. Press, Cambridge, 1989.

    Google Scholar 

  9. [9]

    V. Diekert and A. Weiss: Context-free groups and their structure trees, Internat. J. Algebra Comput.23 (2013), 611–642.

    MathSciNet  Article  Google Scholar 

  10. [10]

    C. Droms, B. Servatius and H. Servatius: The structure of locally finite two-connected graphs, Electron. J. Combin. 2 (1995), Research Paper 17.

  11. [11]

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

    MathSciNet  Article  Google Scholar 

  12. [12]

    M. J. Dunwoody and B. Krön: Vertex cuts, J. Graph Theory80 (2015), 136–171.

    MathSciNet  Article  Google Scholar 

  13. [13]

    P. J. Flory: The configuration of a real polymer chain, J. Chem. Phys.17 (1949), 303–310.

    Article  Google Scholar 

  14. [14]

    H. Freudenthal: Über die Enden diskreter Räume und Gruppen, Comment. Math. Helv.17 (1945), 1–38.

    MathSciNet  Article  Google Scholar 

  15. [15]

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

    MathSciNet  Article  Google Scholar 

  16. [16]

    R. Halin: über unendliche Wege in Graphen, Math. Ann.157 (1964), 125–137.

    MathSciNet  Article  Google Scholar 

  17. [17]

    R. Halin: Automorphisms and endomorphisms of infinite locally finite graphs, Abh. Math. Sem. Univ. Hamburg39 (1973), 251–283.

    MathSciNet  Article  Google Scholar 

  18. [18]

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

    MathSciNet  Article  Google Scholar 

  19. [19]

    M. A. Harrison: Introduction to Formal Language Theory, Addison-Wesley, Reading, MA, 1978.

    MATH  Google Scholar 

  20. [20]

    W. Imrich and N. Seifter: A note on the growth of transitive graphs, Discrete Math.73 (1989), 111–117.

    MathSciNet  Article  Google Scholar 

  21. [21]

    W. Imrich and N. Seifter: A survey on graphs with polynomial growth, Discrete Math.95 (1991), 101–117.

    MathSciNet  Article  Google Scholar 

  22. [22]

    H. A. Jung and M. E. Watkins: Fragments and automorphisms of infinite graphs, European J. Combin.5 (1984), 149–162.

    MathSciNet  Article  Google Scholar 

  23. [23]

    C. Lindorfer: The Language of Self-Avoiding Walks, Springer Spektrum, BestMasters series, 2018.

  24. [24]

    N. Madras and G. Slade: The Self-avoiding Walk, Probability and its Applications, Birkhäuser, Boston, MA, 1993.

    Google Scholar 

  25. [25]

    D. E. Muller and P. E. Schupp: Groups, the theory of ends and context-free languages, J. Comput. System Sc.26 (1983), 295–310.

    MathSciNet  Article  Google Scholar 

  26. [26]

    D. E. Muller and P. E. Schupp: The theory of ends, pushdown automata, and second-order logic, Theoretical Computer Science37 (1985), 51–75.

    MathSciNet  Article  Google Scholar 

  27. [27]

    L. Pélecq: Automorphism groups of context-free graphs, Theoret. Comput. Sci.165 (1996), 275–293.

    MathSciNet  Article  Google Scholar 

  28. [28]

    G. Sabidussi: On a class of fixed-point-free graphs, Proc. Amer. Math. Soc.9 (1958), 800–804.

    MathSciNet  Article  Google Scholar 

  29. [29]

    J.-P. Serre: Trees, Translated from the French by J. Stillwell. Springer-Verlag, Berlin-New York, 1980.

  30. [30]

    C. Thomassen and W. Woess: Vertex-transitive graphs and accessibility, J. Combin. Theory Ser. B58 (1993), 248–268.

    MathSciNet  Article  Google Scholar 

  31. [31]

    V. I. Trofimov: Groups of automorphisms of graphs as topological groups, Mat. Zametki38 (1985), 378–385.

    MathSciNet  Google Scholar 

  32. [32]

    W. T. Tutte: Graph Theory, with a foreword by C. St. J. A. Nash-Williams, Encyclopedia of Mathematics and its Applications 21, Addison-Wesley, Reading, MA, 1984.

    Google Scholar 

  33. [33]

    W. Woess: Context-free pairs of groups. II - cuts, tree sets, and random walks, Discrete Math.312 (2012), 157–173.

    MathSciNet  Article  Google Scholar 

  34. [34]

    W. Woess: Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics 138, Cambridge University Press, Cambridge, 2000.

    Google Scholar 

  35. [35]

    D. Zeilberger: Self-avoiding walks, the language of science, and Fibonacci numbers, J. Statist. Plann. Inference54 (1996), 135–138.

    MathSciNet  Article  Google Scholar 

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Correspondence to Christian Lindorfer.

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This work is partially supported by Austrian Science Fund FWF P31237 and DK W1230.

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Lindorfer, C., Woess, W. The Language of Self-Avoiding Walks. Combinatorica 40, 691–720 (2020). https://doi.org/10.1007/s00493-020-4184-z

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Mathematics Subject Classification (2010)

  • 20F10
  • 68Q45
  • 05C25