Point processes and stochastic displacement fields

TL;DR

This paper derives exact transformation equations for the correlation functions and power spectrum of point processes affected by stochastic displacement fields, analyzing both correlated and uncorrelated cases across dimensions.

Contribution

It provides a general framework with explicit formulas for how stochastic displacements modify point process statistics, including large-scale correlations and superhomogeneity.

Findings

Exact transformation equations for correlation functions and power spectrum.

Analysis of large-scale correlations introduced by displacement fields.

Discussion on the realizability of superhomogeneous point processes.

Abstract

The effect of a stochastic displacement field on a statistically independent point process is analyzed. Stochastic displacement fields can be divided into two large classes: spatially correlated and uncorrelated. For both cases exact transformation equations for the two-point correlation function and the power spectrum of the point process are found, and a detailed study of them with important paradigmatic examples is done. The results are general and in any dimension. A particular attention is devoted to the kind of large scale correlations that can be introduced by the displacement field, and to the realizability of arbitrary ``superhomogeneous'' point processes.