Abstract.
We consider diffraction at random point scatterers on general discrete point sets in ℝν, restricted to a finite volume. We allow for random amplitudes and random dislocations of the scatterers. We investigate the speed of convergence of the random scattering measures applied to an observable towards its mean, when the finite volume tends to infinity. We give an explicit universal large deviation upper bound that is exponential in the number of scatterers. The rate is given in terms of a universal function that depends on the point set only through the minimal distance between points, and on the observable only through a suitable Sobolev-norm. Our proof uses a cluster expansion and also provides a central limit theorem.
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Received: 10 October 2001 / Revised version: 26 January 2003 / Published online: 15 April 2003
Work supported by the DFG
Mathematics Subject Classification (2000): 78A45, 82B44, 60F10, 82B20
Key words or phrases: Diffraction theory – Random scatterers – Random point sets – Quasicrystals – Large deviations – Cluster expansions
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Külske, C. Universal bounds on the selfaveraging of random diffraction measures. Probab. Theory Relat. Fields 126, 29–50 (2003). https://doi.org/10.1007/s00440-003-0261-7
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DOI: https://doi.org/10.1007/s00440-003-0261-7
Keywords
- Limit Theorem
- Minimal Distance
- Central Limit
- Finite Volume
- Central Limit Theorem