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Diffraction of Stochastic Point Sets: Explicitly Computable Examples

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Abstract

Stochastic point processes relevant to the theory of long-range aperiodic order are considered that display diffraction spectra of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. The latter is based on the classical theory of point processes and the Palm distribution. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the Poisson summation formula for lattice Dirac combs.

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References

  1. Ataman Y.: On positive definite measures. Monatsh. Math. 79, 265–272 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baake, M.: Diffraction of weighted lattice subsets. Canad. Math. Bulletin 45, 483–498 (2002), arXiv:math.MG/0106111

    MATH  MathSciNet  Google Scholar 

  3. Baake, M., Frettlöh, D., Grimm, U.: A radial analogue of Poisson’s summation formula with applications to powder diffraction and pinwheel patterns. J. Geom. Phys. 57, 1331–1343 (2007), arXiv:math.SP/ 0610408

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Baake, M., Höffe, M.: Diffraction of random tilings: Some rigorous results. J. Stat. Phys. 99, 219–261 (2000), arXiv:math-ph/9904005

    Article  MATH  Google Scholar 

  5. Baake, M., Lenz, D.: Deformation of Delone dynamical systems and pure point diffraction. J. Fourier Anal. Appl. 11, 125–150 (2005), arXiv:math.DS/0404155

    Article  MATH  MathSciNet  Google Scholar 

  6. Baake, M., Lenz, D., Moody, R.V.: Characterisation of models sets by dynamical systems. Erg. Th. & Dyn. Syst. 27, 341–382 (2007), arXiv:math.DS/0511648

    Article  MATH  MathSciNet  Google Scholar 

  7. Baake, M., Moody, R.V.: Diffractive point sets with entropy. J. Phys. A: Math. Gen. 31, 9023–9038 (1998), arXiv:math-ph/9809002

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. 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

    Article  MATH  MathSciNet  Google Scholar 

  9. Baake, M., Moody, R.V., Pleasants, P.A.B.: Diffraction from visible lattice points and k-th power free integers. Discr. Math. 221, 3–42 (2000), arXiv:math.MG/9906132

    Article  MATH  MathSciNet  Google Scholar 

  10. Baake, M., Sing, B.: Diffraction spectrum of lattice gas models above Tc. Lett. Math. Phys. 68, 165–173 (2004), arXiv:math-ph/0405064

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Baake, M., Zint, N.: Absence of singular continuous diffraction for discrete multi-component particle models. J. Stat. Phys. 130, 727–740 (2008), arXiv:0709.2061(math-ph)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Berberian, S.K.: Measure and Integration. New York: Chelsea, 1965

    MATH  Google Scholar 

  13. Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Berlin: Springer, 1975

    MATH  Google Scholar 

  14. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. New York: Academic Press, 1968

    MATH  Google Scholar 

  15. Bramson M., Cox J.T., Greven A.: Invariant measures of critical spatial branching processes in high dimensions. Ann. Probab. 25, 56–70 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Córdoba A.: Dirac combs. Lett. Math. Phys. 17, 191–196 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. Cowley, J.M.: Diffraction Physics. 3rd ed., Amsterdam: North-Holland, 1995

    Google Scholar 

  18. Daley, D.D., Vere-Jones, D.: An Introduction to the Theory of Point Processes. New York: Springer, 1988

    MATH  Google Scholar 

  19. Daley, D.D., Vere-Jones, D.: An Introduction to the Theory of Point Processes I: Elementary Theory and Methods. 2nd ed., 2nd corr. printing, New York: Springer, 2005

  20. Daley, D.D., Vere-Jones, D.: An Introduction to the Theory of Point Processes II: General Theory and Structure. 2nd ed., New York: Springer, 2008

    MATH  Google Scholar 

  21. Deng, X., Moody, R.V.: Dworkin’s argument revisited: point processes, dynamics, diffraction, and correlations. J. Geom. Phys. 58, 506–541 (2008), arXiv:0712.3287(math.DS)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Dieudonné, J.: Treatise on Analysis. Vol. II, 2nd ed., New York: Academic Press, 1976

    MATH  Google Scholar 

  23. Enter A.C.D., Miȩkisz J.: How should one define a (weak) crystal? J. Stat. Phys. 66, 1147–1153 (1992)

    Article  MATH  ADS  Google Scholar 

  24. Etemadi N.: An elementary proof of the strong law of large numbers. Z. Wahrsch. verw. Gebiete 55, 119–122 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  25. Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed., New York: Wiley, 1972

    Google Scholar 

  26. Gil de Lamadrid, J., Argabright, L.N.: Almost Periodic Measures. Memoirs AMS, Vol. 65, no. 428, Providence, RI: Amer. Math. Soc., 1990

  27. Gnedenko, B.V.: Theory of Probability. 6th ed., Amsterdam: CRC Press, 1998

    Google Scholar 

  28. Goueré, J.-B.: Diffraction and Palm measure of point processes. Comptes Rendus Acad. Sci. (Paris) 342, 141–146 (2003), arXiv:math.PR/0208064

    Google Scholar 

  29. Goueré, J.-B.: Quasicrystals and almost periodicity. Commun. Math. Phys. 255, 655–681 (2005), arXiv:math-ph/0212012

    Article  MATH  ADS  Google Scholar 

  30. Guinier, A.: X-Ray Diffraction in Crystals, Imperfect Crystals and Amorphous Bodies. Reprint, New York: Dover, 1994

  31. Gorostiza L.G., Wakolbinger A.: Persistence criteria for a class of branching particle systems in continuous time. Ann. Probab. 19, 266–288 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  32. Hof A.: On diffraction by aperiodic structures. Commun. Math. Phys. 169, 25–43 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  33. Hof A.: Diffraction by aperiodic structures at high temperatures. J. Phys. A: Math. Gen. 28, 57–62 (1995)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. Höffe, M.: Diffraction of the dart-rhombus random tiling. Mat. Science Eng. 294–296, 373–376 (2000), arXiv:math-ph/9911014

  35. Höffe, M., Baake, M.: Surprises in diffuse scattering. Z. Kristallogr. 215, 441–444 (2000), arXiv:math-ph/0004022

    Article  Google Scholar 

  36. Kallenberg, O.: Random Measures. 3rd ed., Berlin: Akademie-Verlag, 1983

    MATH  Google Scholar 

  37. Karr, A.F.: Point Processes and Their Statistical Inference. 2nd ed., New York: Dekker, 1991

    MATH  Google Scholar 

  38. Kerstan, J., Matthes, K., Mecke, J.: Unbegrenzt teilbare Punktprozesse. Berlin: Akademie-Verlag, 1974

    MATH  Google Scholar 

  39. Kramer P., Neri R.: On periodic and non-periodic space fillings of \({\mathbb{E}^m}\) obtained by projection. Acta Cryst. A40, 580–587 (1984)

    MathSciNet  Google Scholar 

  40. Külske, C.: Universal bounds on the selfaveraging of random diffraction measures. Probab. Th. Rel. Fields 126, 29–50 (2003), arXiv:math-ph/0109005

    Article  MATH  Google Scholar 

  41. Külske C.: Concentration inequalities for functions of Gibbs fields with application to diffraction and random Gibbs measures. Commun. Math. Phys. 239, 29–51 (2003)

    Article  MATH  ADS  Google Scholar 

  42. Lenz D.: Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Commun. Math. Phys. 287, 225–258 (2009), arXiv:math-ph/0608026)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  43. 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)

    MATH  Google Scholar 

  44. Moody, R.V.: Model sets: A survey. In: From Quasicrystals to More Complex Systems, eds. F. Axel, F. Dénoyer, J.P. Gazeau, Les Ulis: EDP Sciences/Berlin: Springer, 2000, pp. 145–166, arXiv:math.MG/ 0002020

  45. Moody R.V.: Uniform distribution in model sets. Can. Math. Bull 45, 123–130 (2002)

    MATH  MathSciNet  Google Scholar 

  46. Nguyen X.X., Zessin H.: Ergodic theorems for spatial processes. Z. Wahrsch. verw. Gebiete 48, 133–158 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  47. Penrose R.: The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271 (1974)

    Google Scholar 

  48. Pinsky, M.A.: Introduction to Fourier Analysis and Wavelets. Pacific Grove, CA: Brooks/Cole, 2002

  49. Radin, C.: Aperiodic tilings, ergodic theory, and rotations. In: The Mathematics of Long-Range Aperiodic Order, ed. R.V. Moody, NATO-ASI Series C 489, Dordrecht: Kluwer, 1997, pp. 499–519

  50. Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. 2nd ed., San Diego: Academic Press, 1980

    MATH  Google Scholar 

  51. Rudin, W.: Real and Complex Analysis. 3rd ed., New York: McGraw Hill, 1987

    MATH  Google Scholar 

  52. Rudin, W.: Fourier Analysis on Groups. reprint, New York: Wiley, 1990

  53. Shechtman D., Blech I., Gratias D., Cahn J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 183–185 (1984)

    Article  Google Scholar 

  54. Schlottmann, M.: Cut-and-project sets in locally compact Abelian groups. In: Quasicrystals and Discrete Geometry. ed. J. Patera, Fields Institute Monographs, Vol. 10, Providence, RI: Amer. Math. Soc., 1998, pp. 247–264

  55. Schlottmann, M.: Generalized model sets and dynamical systems. In: Directions in Mathematical Quasicrystals, eds. M. Baake, R.V. Moody, CRM Monograph Series, Vol. 13, Providence, RI: Amer. Math. Soc., 2000, pp. 143–159

  56. Steurer W. et al.: What is a crystal? Z. Kristallogr. 222, 308–319 (2007)

    Article  Google Scholar 

  57. Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications. Berlin: Akademie-Verlag, 1987

    MATH  Google Scholar 

  58. Stoyan D., Stoyan H.: On one of Matérn’s hard-core point process models. Math. Nachr. 122, 205–214 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  59. Strungaru N.: Almost periodic measures and long-range order in Meyer sets. Discr. Comput. Geom. 33, 483–505 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  60. Urban K., Feuerbacher M.: Structurally complex alloy phases. J. Non-Cryst. Solids 334 & 335, 143–150 (2004)

    Article  Google Scholar 

  61. Ushakov, N.G.: Selected Topics in Characteristic Functions. Utrecht: Brill Academic Publishers, 1999

    MATH  Google Scholar 

  62. Walters, P.: An Introduction to Ergodic Theory. reprint, New York: Springer, 2000

  63. Welberry, T.R.: Diffuse X-Ray Scattering and Models of Disorder. Oxford: Clarendon Press, 2004

    Google Scholar 

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Michael Baake

  2. Institut für Mathematik, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, 55099, Mainz, Germany

    Matthias Birkner

  3. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3P4, Canada

    Robert V. Moody

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  1. Michael Baake
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  2. Matthias Birkner
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  3. Robert V. Moody
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Correspondence to Michael Baake.

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Communicated by H. Spohn

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Baake, M., Birkner, M. & Moody, R.V. Diffraction of Stochastic Point Sets: Explicitly Computable Examples. Commun. Math. Phys. 293, 611–660 (2010). https://doi.org/10.1007/s00220-009-0942-x

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