Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-hyperuniformity of Gibbs point processes with short range interaction

David Dereudre, Daniela Flimmel

Abstract

We investigate the hyperuniformity of marked Gibbs point processes with weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Some variants of stability and range assumptions are posed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models including Gibbs point processes with a superstable, lower-regular, integrable pair potential as well as Widom--Rowlinson model with random radii or Gibbs point processes with interactions based on Voronoi tessellation and nearest neighbour graph.

Non-hyperuniformity of Gibbs point processes with short range interaction

Abstract

We investigate the hyperuniformity of marked Gibbs point processes with weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Some variants of stability and range assumptions are posed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models including Gibbs point processes with a superstable, lower-regular, integrable pair potential as well as Widom--Rowlinson model with random radii or Gibbs point processes with interactions based on Voronoi tessellation and nearest neighbour graph.
Paper Structure (18 sections, 14 theorems, 88 equations)

This paper contains 18 sections, 14 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation
  4. Gibbs point process
  5. Main results
  6. Conditions for (a1), (a2), resp. (A1), (A2)
  7. Stability and range of interaction
  8. Stochastic comparison tools
  9. Examples
  10. Pair potentials
  11. Widom--Rowlinson models
  12. Voronoi interaction
  13. Interactions based on $k$-nearest neighbour graph
  14. Proofs of the main results
  15. Proofs of Theorem \ref{['thm1']} and Theorem \ref{['thm2']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\Gamma$ be a stationary Gibbs point process on $\mathbb{R}^d$ with Papangelou intensity $\lambda^*$ and let the following assumptions are satisfied Then there is a constant $\mathcal{C}_{nhyp}>0$ not depending on $\Lambda$ such that In particular, $\Gamma$ is not hyperuniform.

Theorems & Definitions (47)

  • Definition 1
  • Remark : The form of the Papangelou intensity
  • Remark
  • Theorem 1
  • Theorem 2
  • Remark : Assumptions (A1), (A2)
  • Remark : Shape of $\Lambda$
  • Remark : Unmarked case
  • Definition 2: Local stability
  • Definition 3: Range of interaction
  • ...and 37 more