Journal of Statistical Physics

, Volume 142, Issue 2, pp 223–228 | Cite as

An Ultrametric State Space with a Dense Discrete Overlap Distribution: Paperfolding Sequences

  • Aernout C. D. van EnterEmail author
  • Ellis de Groote
Open Access


We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support (as a closed set) is the full interval [−1,+1]. The space of paperfolding sequences has an ultrametric structure. Our example provides an illustration of some properties which were suggested to occur for pure states in spin glass models.


Pure State Spin Glass Pure Point Cosine Family Replica Symmetry Breaking 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Anagnostopoulou, V., Díaz-Ordaz, K., Jenkinson, O., Richard, C.: Sturmian maximizing measures for the piecewise-linear cosine family. Dresden, preprint (2010) Google Scholar
  2. 2.
    Allouche, J.-P., Mendès France, M.: Automata and automatic sequences. In: Axel, F., Gratias, D. (eds.) Beyond Quasicrystals, pp. 293–367. Les Éditions de Physique/Springer, Berlin (1995) Google Scholar
  3. 3.
    Allouche, J.-P., Shallit, J.: Automatic sequences. Cambridge University Press, Cambridge (2003) zbMATHCrossRefGoogle Scholar
  4. 4.
    Aubry, S.: Weakly periodic structures and example. J. Phys. (Paris) C3-50, 97–106 (1989) MathSciNetGoogle Scholar
  5. 5.
    Baake, M., Höffe, M.: Diffraction of random tilings: Some rigorous results. J. Stat. Phys. 99, 219–261 (2000) zbMATHCrossRefADSGoogle Scholar
  6. 6.
    Baake, M., Moody, R.V., Richard, C., Sing, B.: Which distributions of matter diffract? Some answers. In: Trebin, H.-R. (ed.) Quasicrystals: Structure and Physical Properties, pp. 188–207. Wiley/VCH, New York/Weinheim (2003). arXiv:math-ph/0301019 Google Scholar
  7. 7.
    Baake, M., Moody, R.V., Schlottmann, M.: Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces. J. Phys. A, Math. Gen. 31, 5755–5765 (1998) zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    Baake, M., Moody, R.V.: Wighted Dirac combs with pure point diffraction. J. Reine Angew. Math. 573, 61–94 (2004). arXiv:math.MG/0203030 zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bolthausen, E.: In: Bolthausen, E., Bovier, A. (eds.) Spin Glasses. Lecture Notes in Mathematics, vol. 1900. Springer, Berlin (2007). See in particular p. 16 CrossRefGoogle Scholar
  10. 10.
    Dekking, M., Mendes-France, M., van der Poorten, A.: FOLDS! (I,II and III). Math. Intell. 4, 130–138 (1982), p. 173–181, p. 190–195 zbMATHCrossRefGoogle Scholar
  11. 11.
    van Enter, A.C.D.: On the set of pure states for some systems with non-periodic long-range order. Physica A 232, 600–607 (1996) CrossRefMathSciNetADSGoogle Scholar
  12. 12.
    van Enter, A.C.D., Hof, A., Miękisz, J.: Overlap distributions for deterministic systems with many pure states. J. Phys. A, Math. Gen. 25, L1133–1137 (1992) CrossRefGoogle Scholar
  13. 13.
    van Enter, A.C.D., Miękisz, J.: How should one define a (weak) crystal? J. Stat. Phys. 66, 1147–1153 (1992) zbMATHCrossRefADSGoogle Scholar
  14. 14.
    Gardner, C., Miękisz, J., Radin, C., van Enter, A.C.D.: Fractal symmetry in an Ising model. J. Phys. A, Math. Gen. 22, L1019–1023 (1989) CrossRefADSGoogle Scholar
  15. 15.
    de Groote, E.: The overlap distribution of paperfolding sequences. Groningen bachelor thesis, 2010 Google Scholar
  16. 16.
    Kurchan, J.: Dynamic Heterogeneities in Glasses, Colloids and Granular Matterials, Chap. 1. Oxford University Press, London (2011, to appear). arXiv:1010.2953. From Kurchan’s answer to Question 10: “It is amazing how little attention the glass community has paid to lessons that could be learned from quasicrystals and in general nonperiodic systems.”
  17. 17.
    Lee, J.-Y., Moody, R.V.: Lattice substitution systems and model sets. Discrete Comput. Geom. 25, 173–201 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lee, J.-Y., Moody, R.V., Solomyak, B.: Pure point dynamical and diffraction spectra. Ann. Inst. Henri Poincaré, Theor. Math. Phys. 3, 1003–1018 (2002) zbMATHMathSciNetGoogle Scholar
  19. 19.
    Lü, K., Wang, J.: Construction of sturmian sequences. J. Phys. A, Math. Gen. 38, 2891–2897 (2005) CrossRefADSGoogle Scholar
  20. 20.
    Mézard, M., Parisi, G., Virasoro, M.A.: Spin Glass Theory and Beyond. Singapore, World Scientific (1987) zbMATHGoogle Scholar
  21. 21.
    Miękisz, J.: Quasicrystals: Microscopic models of non-periodic structures. In: Leuven Lecture Notes in Mathematical and Theoretical Physics, vol. 5 (1993) Google Scholar
  22. 22.
    Newman, C.M.: Topics in Disordered Systems. ETH Lectures in Mathematics. Birkhäuser, Basel/Boston/Berlin (1997) zbMATHCrossRefGoogle Scholar
  23. 23.
    Newman, C.M., Stein, D.L.: The metastate approach to thermodynamic chaos. Phys. Rev. E 55, 594–5211 (1997) MathSciNetGoogle Scholar
  24. 24.
    Newman, C.M., Stein, D.L.: Thermodynamic chaos and the structure of short-range spin glasses. In: Bovier, A., Picco, P. (eds.) Mathematical Aspects of Spin Glasses and Neural Networks, pp. 243–287. Birkhäuser, Boston/Basel/Berlin (1998) CrossRefGoogle Scholar
  25. 25.
    Newman, C.M., Stein, D.L.: Ordering and broken symmetry in short-ranged spin glasses. J. Phys., Condens. Matter 15, R1319–R1364 (2003). arXiv:cond-mat/0301403, Topical Review CrossRefADSGoogle Scholar
  26. 26.
    Newman, C.M., Stein, D.L.: The state(s) of replica symmetry breaking: Mean field theories versus short-ranged spin glasses. (Formerly known as “Replica Symmetry Breaking’s New Clothes”). J. Stat. Phys. 106, 213–244 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Newman, C.M., Stein, D.L.: Distribution of pure states in short-range spin glasses. Int. J. Mod. Phys. B24, 2091–2106 (2010) MathSciNetADSGoogle Scholar
  28. 28.
    Parisi, G., Sourlas, N.: P-adic numbers and replica symmetry breaking. Eur. J. Phys. B14, 235–242 (2000) MathSciNetGoogle Scholar
  29. 29.
    Parisi, G., Talagrand, M.: On the distribution of the overlaps at given disorder. C. R. Acad. Sci., Ser. I Math. 339, 303–306 (2004) zbMATHMathSciNetGoogle Scholar
  30. 30.
    Radin, C.: Disordered ground states of classical lattice models. Rev. Math. Phys. 3, 125–135 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Radin, C.: Low temperature and the origin of crystalline symmetry. Int. J. Mod. Phys. B 1, 1157–1191 (1987) CrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Ruelle, D.: Do turbulent crystals exist? Physica 113A, 619–623 (1982) MathSciNetADSGoogle Scholar
  33. 33.
    Schechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translation symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984) CrossRefADSGoogle Scholar
  34. 34.
    Schlottmann, M.: Generalized model sets and dynamical systems. In: Baake, M., Moody, R.V. (eds.) Directions in Mathematical Quasicrystals. CRM Monograph Series, vol. 13, pp. 143–159. AMS, Providence (2000) Google Scholar
  35. 35.
    Senechal, M.: Quasicrystals and Geometry. Cambridge University Press, Cambridge (1995) zbMATHGoogle Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Johann Bernoulli InstituteRijksuniversiteit GroningenGroningenThe Netherlands
  2. 2.BeilenThe Netherlands

Personalised recommendations