Universal bounds on the selfaveraging of random diffraction measures

TL;DR

This paper establishes universal bounds on how quickly random diffraction measures converge to their mean in large volumes, using probabilistic and analytical techniques, applicable to various random scatterer configurations.

Contribution

It introduces a universal large deviation bound and a central limit theorem for diffraction measures with random scatterers, depending only on minimal point distance and Sobolev norms.

Findings

Derived exponential large deviation bounds for diffraction measures.

Proved a central limit theorem for the convergence of measures.

Bounded convergence speed in terms of minimal point distance and Sobolev norms.

Abstract

We consider diffraction at random point scatterers on general discrete point sets in $\R^\nu$, 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.