[Paper Review] Regularization of a stationary point process by a stationary increments perturbation

Loïc Thomassey, Raphaël Lachièze-Rey+1 more|arXiv (Cornell University)|Feb 23, 2026

Point processes and geometric inequalities0 citations

TL;DR

The paper introduces a Palm-distribution-based perturbation of stationary point processes using fractional Brownian fields to regularize lattice structures, yielding hyperuniform processes with efficient n log n point generation in 1D and an explicit structure factor formula.

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).

Motivation & Objective

• Motivate and construct regularized stationary point processes by perturbing Palm distributions of lattices.
• Show that perturbation with a d-dimensional fractional Brownian field yields a Palm-distributed ergodic process with absolutely continuous Bartlett’s spectrum.
• Derive a tractable expression for the structure factor of the perturbed process and discuss hyperuniformity properties, especially in 1D.
• Demonstrate computationally efficient generation of hyperuniform point sets in dimension 1 using fractional Brownian motion.

Proposed method

• Define perturbed Palm lattice by decorating Palm distribution  with a stationary increments Gaussian process B.
• Prove that _B = {x + B_x : x in } is the Palm distribution of an ergodic stationary point process _B.
• Show that the Bartlett spectrum of _B is absolutely continuous with density s_{_B}(t) = E[ -integral of e^{-1/2 (Σ_x t, t)} e^{-i(t, x)} (dx) ] for t ≠ 0.
• Specialize to d-fBf with independent coordinates and a variogram Σ_t, yielding explicit structure-factor formula.
• Discuss mixing, ergodicity, and hyperuniformity implications, including asymptotics in 1D with fractional Brownian motion (fBm).
• Provide numerical simulation notes showing n log n generation efficiency for 1D perturbed Palm lattice via Gaussian increments.

Experimental results

Research questions

• RQ1 Can perturbing the Palm distribution of a stationary point process with a d-dimensional fractional Brownian field erase lattice periodicity and yield a hyperuniform stationary process?
• RQ2 What is the explicit structure factor of the perturbed process and under what conditions is it absolutely continuous (no atomic part)?
• RQ3 Do perturbations preserve ergodicity and lead to mixing properties for the resulting process?
• RQ4 How does the hyperuniformity behavior depend on the Hurst indices in 1D, and what are the variance-growth implications?
• RQ5 Is there a computationally efficient method to sample the perturbed Palm lattice in 1D for large n?

Key findings

• The perturbed Palm lattice _B is the Palm distribution of a stationary ergodic point process with the same intensity as the original.
• The structure factor s_{_B}(t) is absolutely continuous for t ≠ 0 and given by an expectation involving the variogram Σ_x and the Palm measure, ensuring removal of lattice atomic components.
• In 1D, with B as fractional Brownian motion of index h, the structure factor behaves as |t|^{1-2h} near zero, implying hyperuniformity for h < 1/2.
• The lattice structure is erased by the fBm fluctuations, eliminating the atomic part of the spectrum and producing a hyperuniform process with a variance growth scaling related to the Hurst parameter.
• Numerical simulations indicate computational efficiency: sampling n points of a Gaussian process with stationary increments can be done in n log(n) operations in 1D, enabling large-scale generation of hyperuniform configurations.