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

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.

💡 Research Summary

This paper introduces a novel class of stochastic models called Gibbs perturbed lattice (GPL) models, which sit at the intersection of classical Gibbs lattice systems, continuous Gibbs point processes, and hyperuniform point processes. The authors start by highlighting a gap in the literature: traditional Gibbs lattice models are flexible but generally not hyperuniform, while known hyperuniform models such as the Ginibre process or simple perturbed lattices lack the ability to enforce physically relevant constraints like hard‑core repulsion. To bridge this gap, they consider a full‑rank lattice (L\subset\mathbb{R}^d) and associate to each lattice site (i) an independent displacement (or “move”) (X_i) drawn from a reference distribution (Q). The perturbed point set is (\Gamma={i+X_i+U: i\in L}), where (U) is a uniform random shift that restores translation invariance. The key novelty lies in defining a Hamiltonian (H) that depends only on the actual positions of the perturbed points, not on the marks themselves. This allows the model to retain a one‑to‑one correspondence between lattice sites and points while introducing interaction through the geometry of the point cloud.

The paper establishes a rigorous mathematical foundation for GPL models. Under a set of technical assumptions on the Hamiltonian—stability (H1), non‑degeneracy (H2), heredity (H3), translation invariance (H4), and either finite‑range or distance‑dependent pairwise structure (H5)—and on the move distribution—positivity on a small ball (Q0), finite exponential moments (Q1), bounded support (Q2), and finite first moment (Q3)—the authors prove the existence of finite‑volume Gibbs measures (\mu_\Lambda) and, via Dobrushin–Lanford–Ruelle (DLR) theory, the existence of an infinite‑volume stationary Gibbs measure (\mu). They derive first‑order, second‑order, and variational DLR equations that play the role of GNZ equations for this hybrid setting. Ergodicity and translation invariance of (\mu) are also shown.

A central contribution is the analysis of hyperuniformity. Building on results for i.i.d. perturbed lattices, the authors demonstrate that when the Hamiltonian enforces sufficient repulsion (e.g., hard‑core Strauss interaction or Lennard‑Jones type potentials) and the move distribution has a light enough tail, the variance of the number of points in a large ball grows slower than the volume, i.e. (\operatorname{Var}