Diffraction and Palm measure of point processes

TL;DR

This paper establishes the existence of diffraction measures for all stationary and ergodic point processes using Palm measures, providing explicit formulas for certain classes like cut-and-project sets.

Contribution

It introduces a general proof of diffraction measure existence for stationary ergodic point processes and derives explicit expressions for specific process classes.

Findings

Diffraction measure exists for all stationary and ergodic point processes.

Explicit formulas are derived for cut-and-project sets.

Results apply to stochastic subsets of Z^d.

Abstract

Using the Palm measure notion, we prove the existence of the diffraction measure of all stationary and ergodic point processes. We get precise expressions of those measures in the case of specific processes : stochastic subsets of Z^d, sets obtained by the ``cut-and-project'' method.