Journal of Statistical Physics

, Volume 143, Issue 1, pp 88–101 | Cite as

Close-packed Dimers on the Line: Diffraction versus Dynamical Spectrum

  • Michael BaakeEmail author
  • Aernout van Enter
Open Access


The translation action of ℝ d on a translation bounded measure ω leads to an interesting class of dynamical systems, with a rather rich spectral theory. In general, the diffraction spectrum of ω, which is the carrier of the diffraction measure, lives on a subset of the dynamical spectrum. It is known that, under some mild assumptions, a pure point diffraction spectrum implies a pure point dynamical spectrum (the opposite implication always being true). For other systems, the diffraction spectrum can be a proper subset of the dynamical spectrum, as was pointed out for the Thue-Morse sequence (with singular continuous diffraction) by van Enter and Miȩkisz (J. Stat. Phys. 66:1147–1153, 1992). Here, we construct a random system of close-packed dimers on the line that have some underlying long-range periodic order as well, and display the same type of phenomenon for a system with absolutely continuous spectrum. An interpretation in terms of ‘atomic’ versus ‘molecular’ spectrum suggests a way to come to a more general correspondence between these two types of spectra.


Dynamical systems Diffraction spectra Dynamical spectra Random systems 


  1. 1.
    Aaronson, J., Gilat, D., Keane, M., de Valk, V.: An algebraic construction of a class of one-dependent processes. Ann. Probab. 17, 128–143 (1989) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Baake, M.: Diffraction of weighted lattice subsets. Can. Math. Bull. 45, 483–498 (2002). arXiv:math.MG/0106111 CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baake, M., Birkner, M., Moody, R.V.: Diffraction of stochastic point sets: Explicitly computable examples. Commun. Math. Phys. 293, 611–660 (2009). arXiv:0803.1266 CrossRefMathSciNetGoogle Scholar
  4. 4.
    Baake, M., van Enter, A.C.D., Lenz, D.: On the relation between dynamical and diffraction spectra, in preparation Google Scholar
  5. 5.
    Baake, M., Grimm, U.: The singular continuous diffraction measure of the Thue-Morse chain. J. Phys. A, Math. Theor. 41, 422001 (2008). arXiv:0809.0580 CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Baake, M., Grimm, U.: Kinematic diffraction is insufficient to distinguish order from disorder. Phys. Rev. B 79, 020203(R) (2009) and Phys. Rev. B 80, 029903(E) (2009). arXiv:0810.5750 CrossRefADSGoogle Scholar
  7. 7.
    Baake, M., Grimm, U.: Diffraction of limit periodic point sets. Philos. Mag. (2010). doi: 10.1080/14786435.2010.508447. arXiv:1007.0707
  8. 8.
    Baake, M., Lenz, D.: Dynamical systems on translation bounded measures: pure point dynamical and diffraction spectra. Ergod. Theory Dyn. Syst. 24, 1867–1893 (2004). math.DS/0302231 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Baake, M., Lenz, D.: Deformation of Delone dynamical systems and topological conjugacy. J. Fourier Anal. Appl. 11, 125–150 (2005). math.DS/0404155 CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Baake, M., Lenz, D., Moody, R.V.: Characterization of model sets by dynamical systems. Ergod. Theory Dyn. Syst. 27, 341–382 (2007). arXiv:math/0511648 CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Baake, M., Moody, R.V.: Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. (Crelle) 573, 61–94 (2004). arXiv:math.MG/0203030 CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    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). math-ph/9901008 CrossRefzbMATHADSMathSciNetGoogle Scholar
  13. 13.
    Baake, M., Sing, B.: Diffraction spectrum of lattices gas models above T c. Lett. Math. Phys. 68, 165–173 (2004). math-ph/0405064 CrossRefzbMATHADSMathSciNetGoogle Scholar
  14. 14.
    Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Springer, Berlin (1975) zbMATHGoogle Scholar
  15. 15.
    Cowley, J.M.: Diffraction Physics, 3rd edn. North-Holland, Amsterdam (1995) Google Scholar
  16. 16.
    Denker, M., Grillenberger, C., Sigmund, K.: Ergodic Theory of Compact Spaces. LNM, vol. 527. Springer, Berlin (1976) Google Scholar
  17. 17.
    van Enter, A.C.D., Miȩkisz, J.: How should one define a (weak) crystal? J. Stat. Phys. 66, 1147–1153 (1992) CrossRefzbMATHADSGoogle Scholar
  18. 18.
    Etemadi, N.: An elementary proof of the strong law of large numbers. Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119–122 (1981) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gil de Lamadrid, J., Argabright, L.N.: Almost Periodic Measures. Memoirs AMS, vol. 85(428). AMS, Providence (1990) Google Scholar
  20. 20.
    Goueré, J.-B.: Diffraction and Palm measure of point processes. C. R. Acad. Sci. (Paris) 342, 141–146 (2003). arXiv:math.PR/0208064 Google Scholar
  21. 21.
    Hof, A.: On diffraction by aperiodic structures. Commun. Math. Phys. 169, 25–43 (1995) CrossRefzbMATHADSMathSciNetGoogle Scholar
  22. 22.
    Höffe, M., Baake, M.: Surprises in diffuse scattering. Z. Kristallogr. 215, 441–444 (2000). arXiv:math-ph/0004022 CrossRefGoogle Scholar
  23. 23.
    Külske, C.: Universal bounds on the selfaveraging of random diffraction measures. Probab. Theory Relat. Fields 126, 29–50 (2003). arXiv:math-ph/0109005 CrossRefzbMATHGoogle Scholar
  24. 24.
    Külske, C.: Concentration inequalities for functions of Gibbs fields with application to diffraction and random Gibbs measures. Commun. Math. Phys. 239, 29–51 (2003) CrossRefzbMATHADSGoogle Scholar
  25. 25.
    Lee, J.-Y., Moody, R.V., Solomyak, B.: Pure point dynamical and diffraction spectra. Ann. Henri Poincaré 3, 1003–1018 (2002). arXiv:0910.4809 CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Lenz, D., Strungaru, N.: Pure point spectrum for measure dynamical systems on locally compact Abelian groups. J. Math. Pures Appl. 92, 323–341 (2009). arXiv:0704.2498 CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Queffélec, M.: Substitution Dynamical Systems—Spectral Analysis, 2nd edn. Springer, Berlin (2010) CrossRefzbMATHGoogle Scholar
  28. 28.
    Robinson, E.A. Jr.: Symbolic dynamics and tilings of ℝd. Proc. Symp. Appl. Math. 60, 81–119 (2004) Google Scholar
  29. 29.
    Rudin, W.: Some theorems on Fourier coefficients. Proc. Am. Math. Soc. 10, 855–859 (1959) CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Schlottmann, M.: Generalised 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
  31. 31.
    Shapiro, H.: Extremal problems for polynomials and power series. Masters Thesis, MIT, Boston (1951) Google Scholar
  32. 32.
    Slawny, J.: Ergodic properties of equilibrium states. Commun. Math. Phys. 80, 477–483 (1981) CrossRefADSMathSciNetGoogle Scholar
  33. 33.
    Walters, P.: An Introduction to Ergodic Theory. Springer, New York (2000). Reprint zbMATHGoogle Scholar
  34. 34.
    Withers, R.L.: Disorder, structured diffuse scattering and the transmission electron microscope. Z. Kristallogr. 220, 1027–1034 (2005) Google Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands

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