Random Fields on Model Sets with Localized Dependency and Their Diffraction

Yohji Akama, Shinji Iizuka

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)478-495
Number of pages18
JournalJournal of Statistical Physics
Volume149
Issue number3
DOIs
Publication statusPublished - 2012 Nov

Keywords

  • Absolutely continuous spectrum
  • Diffraction
  • Model set
  • Pure-point spectrum
  • Quasicrystal

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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