Regularization of a stationary point process by a stationary increments perturbation

TL;DR

This work introduces a regularization mechanism for stationary point processes by convolving with a stationary-increments field, effectively removing lattice-like spectral features and enabling hyperuniform configurations. By focusing on Palm distributions and fractional Brownian fields, the authors derive conditions under which the perturbed Palm lattice yields a stationary ergodic process with an explicitly computable structure factor. In 1D, they establish a precise hyperuniformity criterion tied to the Hurst index $h$, with $h<\tfrac{1}{2}$ producing hyperuniformity and a $r^{2h}$ growth of number variance, while $h=\tfrac{1}{2}$ (Brownian) and $h>\tfrac{1}{2}$ exhibit non-hyperuniform behavior. The theoretical results are complemented by numerical simulations demonstrating efficient $O(n\log n)$ generation and de-Palmization strategies, and the work outlines open questions on mixing and rigidity properties of the perturbed Palm construction.

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).

Figures (1)

• Figure 1: Log-regression of hyperuniformity exponent of $\xi_{B}$.

Theorems & Definitions (38)

• Theorem 1.1
• Remark 1.2
• Proposition 1.3
• Theorem 2.1
• Corollary 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3