×

Random fields on model sets with localized dependency and their diffraction. (English) Zbl 1259.82116

Summary: For a random field on a general discrete set, we introduce a condition that the range of the correlation from each site is within a predefined compact set \(D\). For such a random field \(\omega\) defined on the model set \(\Lambda\) that satisfies a natural geometric condition, we develop a method to calculate the diffraction measure of the random field. The method partitions the random field into a finite number of random fields, each being independent and admitting the law of large numbers. The diffraction measure of \(\omega\) consists almost surely of a pure-point component and an absolutely continuous component. The former is the diffraction measure of the expectation \(E[\omega]\), while the inverse Fourier transform of the absolutely continuous component of \(\omega\) turns out to be a weighted Dirac comb which satisfies a simple formula. Moreover, the pure-point component will be understood quantitatively in a simple exact formula if the weights are continuous over the internal space of \(\Lambda\). Then we provide a sufficient condition that the diffraction measure of a random field on a model set is still pure-point.

MSC:

82D25 Statistical mechanics of crystals
82D20 Statistical mechanics of solids
74E15 Crystalline structure
60G60 Random fields
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Argabright, L., Gil de Lamadrid, J.: Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups. Memoirs of the American Mathematical Society, vol. 145. Am. Math. Soc., Providence (1974) · Zbl 0294.43002
[2] Baake, M., Birkner, M., Moody, R.V.: Diffraction of stochastic point sets: explicitly computable examples. Commun. Math. Phys. 293, 611–660 (2010) · Zbl 1197.82053 · doi:10.1007/s00220-009-0942-x
[3] Baake, M., Hermisson, J., Pleasants, P.A.B.: The torus parametrization of quasiperiodic LI-classes. J. Phys. A, Math. Gen. 30(9), 3029–3056 (1997) · Zbl 0919.52015 · doi:10.1088/0305-4470/30/9/016
[4] Baake, M., Lenz, D., Moody, R.V.: Characterization of model sets by dynamical systems. Ergod. Theory Dyn. Syst. 27(2), 341–382 (2007). doi: 10.1017/S0143385706000800 · Zbl 1114.82022 · doi:10.1017/S0143385706000800
[5] Baake, M., Moody, R.V.: Diffractive point sets with entropy. J. Phys. A, Math. Gen. 31, 9023–9039 (1998) · Zbl 0954.82009 · doi:10.1088/0305-4470/31/45/003
[6] Baake, M., Moody, R.V.: Self-similar measures for quasicrystals. In: Directions in Mathematical Quasicrystals. CRM Monogr. Ser., vol. 13, pp. 1–42. Am. Math. Soc., Providence (2000) · Zbl 0972.52013
[7] Baake, M., Moody, R.V.: Weighted Dirac combs with pure point diffraction. J. Reine Angew. Math. 573, 61–94 (2004) · Zbl 1188.43008
[8] 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(27), 5755–5765 (1998). doi: 10.1088/0305-4470/31/27/006 · Zbl 0910.52018 · doi:10.1088/0305-4470/31/27/006
[9] Baake, M., Zint, N.: Absence of singular continuous diffraction for discrete multi-component particle models. J. Stat. Phys. 130(4), 727–740 (2008). doi: 10.1007/s10955-007-9445-3 · Zbl 1214.82113 · doi:10.1007/s10955-007-9445-3
[10] Barabash, R., Ice, G., Turchi, P. (eds.): Diffuse Scattering and the Fundamental Properties of Materials. Momentum Press, New York (2009)
[11] Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis: vol. I. Structure of Topological Groups, Integration Theory, Group Representations, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115. Springer, Berlin (1979) · Zbl 0416.43001
[12] Hof, A.: Diffraction by aperiodic structures at high temperatures. J. Phys. A, Math. Gen. 28(1), 57–62 (1995) · Zbl 0853.60090 · doi:10.1088/0305-4470/28/1/012
[13] Hof, A.: On diffraction by aperiodic structures. Commun. Math. Phys. 169(1), 25–43 (1995) · Zbl 0821.60099 · doi:10.1007/BF02101595
[14] Janot, C.: Quasicrystals: A Primer. Monographs on the Physics and Chemistry of Materials, vol. 50, 2nd edn. Oxford University Press, Oxford (1994) · Zbl 0838.52023
[15] Külske, C.: Concentration inequalities for functions of Gibbs fields with application to diffraction and random Gibbs measures. Commun. Math. Phys. 239(1–2), 29–51 (2003) · Zbl 1023.60080 · doi:10.1007/s00220-003-0841-5
[16] Lenz, D.: Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks. Commun. Math. Phys. 287(1), 225–258 (2009). doi: 10.1007/s00220-008-0594-2 · Zbl 1178.37011 · doi:10.1007/s00220-008-0594-2
[17] Lenz, D., Richard, C.: Pure point diffraction and cut and project schemes for measures: the smooth case. Math. Z. 256(2), 347–378 (2007). doi: 10.1007/s00209-006-0077-0 · Zbl 1129.28003 · doi:10.1007/s00209-006-0077-0
[18] Meyer, Y.: Algebraic Numbers and Harmonic Analysis. North-Holland Mathematical Library, vol. 2. North-Holland, Amsterdam (1972) · Zbl 0267.43001
[19] Moody, R.V.: Meyer sets and their duals. In: The Mathematics of Long-Range Aperiodic Order, Waterloo, ON, 1995. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 489, pp. 403–441. Kluwer Academic, Dordrecht (1997) · Zbl 0880.43008
[20] Moody, R.V.: Uniform distribution in model sets. Can. Math. Bull. 45(1), 123–130 (2002) · Zbl 1007.52013 · doi:10.4153/CMB-2002-015-3
[21] Müller, P., Richard, C.: Ergodic properties of randomly coloured point sets. Can. J. Math. (2012). doi: 10.4153/CJM-2012-009-7 · Zbl 1351.37030
[22] Nevo, A.: Pointwise ergodic theorems for actions of groups. In: Handbook of Dynamical Systems, vol. 1B, pp. 871–982. Elsevier, Amsterdam (2006) · Zbl 1130.37310
[23] Schlottmann, M.: Generalized model sets and dynamical systems. In: Directions in Mathematical Quasicrystals. CRM Monogr. Ser., vol. 13, pp. 143–159. Am. Math. Soc., Providence (2000) · Zbl 0984.37005
[24] Stroock, D.W.: Probability Theory: An Analytic View. Cambridge University Press, Cambridge (1993) · Zbl 0925.60004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.