Regularization of a stationary point process by a stationary increments perturbation

TL;DR

This paper introduces a new regularization method for stationary point processes using convolution with a stationary increments random field, effectively reducing dependencies and eliminating spectral peaks, demonstrated through efficient hyperuniform process generation.

Contribution

The paper proposes a novel regularization technique for stationary point processes via convolution with stationary increments fields, enabling efficient hyperuniform process generation.

Findings

Reduces dependency between distant points in the process.

Erases spectral peaks associated with periodicity.

Efficiently generates hyperuniform point processes in one dimension.

Abstract

We present a novel procedure where a stationary point process is regularized through the convolution with a continuous random field with stationary increments, in the sense that the dependency between distant points is weakened; and the potential peaks in the spectrum (or Bragg peaks), reminiscent of a periodic behavior, are erased. We use this procedure to efficiently generate a hyperuniform point process in dimension 1 using a fractional Brownian Motion; simulating n points with complexity n log(n).