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

TL;DR

This paper introduces Gibbs perturbed lattice models that combine lattice structure with hyperuniformity, establishing their existence, properties, and statistical inference methods, thus bridging a gap between classical Gibbs models and hyperuniform point processes.

Contribution

It proposes a new class of Gibbs perturbed lattice models, proves their existence and hyperuniformity, and develops asymptotically consistent statistical inference methods.

Findings

Existence of Gibbs measures for the proposed models

Some models exhibit hyperuniformity

Development of asymptotic inference procedures

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.