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Existence, properties, and parametric inference for possibly hyperuniform Gibbs perturbed lattices

Jean-François Coeurjolly, Christopher Renaud-Chan

Abstract

This work lies at the intersection of Gibbs models and hyperuniform point processes. Classical Gibbs models, whether defined on lattices or in continuous space, provide flexible tools to describe interacting particle systems but are generally not hyperuniform. Conversely, known hyperuniform models such as the Ginibre process or perturbed lattices lack flexibility and typically cannot enforce physically relevant constraints such as hard-core interactions. We introduce a new class of models, termed Gibbs perturbed lattice models, which preserve a lattice structure while allowing interactions through a Hamiltonian defined on the perturbed particle locations. We establish existence results for the associated Gibbs measures, derive DLR-type equilibrium equations, and show that some models in this class exhibit hyperuniformity. Finally, we propose statistical inference methods based on the Takacs-Fiksel type approach and prove their asymptotic properties.

Existence, properties, and parametric inference for possibly hyperuniform Gibbs perturbed lattices

Abstract

This work lies at the intersection of Gibbs models and hyperuniform point processes. Classical Gibbs models, whether defined on lattices or in continuous space, provide flexible tools to describe interacting particle systems but are generally not hyperuniform. Conversely, known hyperuniform models such as the Ginibre process or perturbed lattices lack flexibility and typically cannot enforce physically relevant constraints such as hard-core interactions. We introduce a new class of models, termed Gibbs perturbed lattice models, which preserve a lattice structure while allowing interactions through a Hamiltonian defined on the perturbed particle locations. We establish existence results for the associated Gibbs measures, derive DLR-type equilibrium equations, and show that some models in this class exhibit hyperuniformity. Finally, we propose statistical inference methods based on the Takacs-Fiksel type approach and prove their asymptotic properties.
Paper Structure (49 sections, 13 theorems, 164 equations, 2 figures, 1 table)

This paper contains 49 sections, 13 theorems, 164 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Lattice, perturbed lattices and point processes
  4. Hamiltonian of a system and move distribution
  5. Hamiltonian or energy functional
  6. Move (or mark) distribution
  7. Existence and properties of Gibbs perturbed lattices
  8. Gibbs measure on a finite sub-lattice
  9. Infinite Gibbs perturbed lattice
  10. Hyperuniformity of Gibbs perturbed lattices
  11. DLR type equations
  12. Parametric inference of Gibbs perturbed lattices
  13. Notation and parametric model
  14. Statistical context and methodologies
  15. Asymptotic results
  16. ...and 34 more sections

Key Result

Proposition 3.3

Let $\Delta \subset \Lambda \subset \mathcal{L}$ be finite, we have for $\mathbb{P}_\Lambda$-a.e $\mathbf{x}_{\Lambda \setminus \Delta}$, where $Z_\Delta(\mathbf{x}_{\Lambda \setminus \Delta})$ is the normalizing constant given by

Figures (2)

  • Figure 1: Illustration of the two statistical frameworks considered in this paper. In (a)-(b) the inside grey rectangle corresponds to $W_n$, crosses to points $i\in \mathcal{L}_U$ and dots to $i+x_i$, for $i\in \mathcal{L}_U$. For (a) Framework 1, we observe crosses ($i$) and dots ($i+x_i$) that fall in $W_n$. For (b) Framework 2, we observe only dots falling in $W_n$.
  • Figure 2: Simulated Gibbs perturbed lattices with Strauss interaction and Gaussian moves.

Theorems & Definitions (29)

  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3: DLR Equations
  • proof
  • Definition 3.4
  • Definition 3.5
  • Theorem 1
  • Proposition 3.6
  • Proposition 3.7
  • Proposition 3.8
  • ...and 19 more