TY - JOUR

T1 - Random Fields on Model Sets with Localized Dependency and Their Diffraction

AU - Akama, Yohji

AU - Iizuka, Shinji

PY - 2012/11

Y1 - 2012/11

N2 - 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 ω defined on the model set Λ 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 ω consists almost surely of a pure-point component and an absolutely continuous component. The former is the diffraction measure of the expectation E[ω], while the inverse Fourier transform of the absolutely continuous component of ω 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 Λ. Then we provide a sufficient condition that the diffraction measure of a random field on a model set is still pure-point.

AB - 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 ω defined on the model set Λ 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 ω consists almost surely of a pure-point component and an absolutely continuous component. The former is the diffraction measure of the expectation E[ω], while the inverse Fourier transform of the absolutely continuous component of ω 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 Λ. Then we provide a sufficient condition that the diffraction measure of a random field on a model set is still pure-point.

KW - Absolutely continuous spectrum

KW - Diffraction

KW - Model set

KW - Pure-point spectrum

KW - Quasicrystal

UR - http://www.scopus.com/inward/record.url?scp=84867996750&partnerID=8YFLogxK

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U2 - 10.1007/s10955-012-0588-5

DO - 10.1007/s10955-012-0588-5

M3 - Article

AN - SCOPUS:84867996750

SN - 0022-4715

VL - 149

SP - 478

EP - 495

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 3

ER -